Extremely high-intensity laser interactions with fundamental quantum

(Dated: April 26, 2012)

The field of laser-matter interaction traditionally deals with the response of atoms,arXiv:1111.3886v2 [hep-ph] 25 Apr 2012

molecules and plasmas to an external light wave. However, the recent sustained tech- nological progress is opening up the possibility of employing intense laser radiation to trigger or substantially influence physical processes beyond atomic-physics energy scales. Available optical laser intensities exceeding 1022 W/cm2 can push the fundamental light- electron interaction to the extreme limit where radiation-reaction effects dominate the electron dynamics, can shed light on the structure of the quantum vacuum, and can trigger the creation of particles like electrons, muons and pions and their corresponding antiparticles. Also, novel sources of intense coherent high-energy photons and laser- based particle colliders can pave the way to nuclear quantum optics and may even allow for potential discovery of new particles beyond the Standard Model. These are the main topics of the present article, which is devoted to a review of recent investigations on high-energy processes within the realm of relativistic quantum dynamics, quantum electrodynamics, nuclear and particle physics, occurring in extremely intense laser fields.

A. Electroweak sector of the Standard Model 47 the control of high-harmonic spectra by spatio-temporal B. Particle physics beyond the Standard Model 47 shaping of the driving pulse (Winterfeldt et al., 2008), onXIV. Conclusion and outlook 49 harmonic generation in laser-plasma interaction (Teub- ner and Gibbon, 2009) and on the dynamics of clusters Acknowledgments 50 in strong laser fields (Fennel et al., 2010). By increasing the optical laser intensity to the or- List of frequently-used symbols 50 der of 1017 -1018 W/cm2 , another physically important References 50 regime in laser-matter interaction is entered: the rel- ativistic regime. In such intense electromagnetic fields an electron reaches relativistic velocities already withinI. INTRODUCTION one laser period, the magnetic component of the Lorentz force becomes of the same order of magnitude of the elec- The first realization of the laser in 1960 (Maiman, tric one, and the electron’s motion becomes highly non-1960) is one of the most important technological break- linear as a function of the laser’s electromagnetic field.throughs. Nowadays lasers are indispensable tools for Although the increasing influence of the magnetic forceinvestigating physical processes in different areas rang- causes a suppression of atomic HHG in the relativistic do-ing from atomic and plasma physics to nuclear and high- main, the highly nonlinear motion of the electrons in suchenergy physics. This has been possible mainly due to the strong laser fields is at the origin of numerous new effectscontinuous progress made along two specific directions: as relativistic self-focusing in plasma and laser wakefielddecrease of the laser pulse duration and increase of the acceleration (Mulser and Bauer, 2010). In the recent re-laser peak intensity (Mourou and Tajima, 2011). On the views Ehlotzky et al., 2009; Mourou et al., 2006; andone hand, multiterawatt laser systems with a pulse du- Salamin et al., 2006, different processes occurring at rel-ration in the femtosecond time scale are readily available ativistic laser intensities are discussed. In particular, innowadays and different laboratories have succeeded in the Ehlotzky et al., 2009, QED processes like Compton, Mottgeneration of single attosecond pulses. Physics at the at- and Møller scattering in a strong laser field are covered, intosecond time scale has been the subject of the recent Mourou et al., 2006, technical aspects and new possibili-review Krausz and Ivanov, 2009. In this review it has ties of the CPA techniques are reviewed together with rel-been pointed out how pulses in the attosecond domain ativistic effects in laser-plasma interaction as, for exam-allow for the detailed investigation of the electron mo- ple, self-induced transparency and wakefield generation,tion in atoms and during molecular reactions. The pro- while in Salamin et al., 2006, spin-effects as well as rel-duction of ultrashort pulses is strongly connected with ativistic multiphoton and tunneling recollision dynamicsthe increase of the laser peak intensity. This is not only in laser-atom interactions are reviewed. Also in the samebecause temporal compression evidently implies an in- year another review was published on nonlinear collectivecrease in intensity at a given laser energy, but also be- photon interactions, including vacuum-polarization ef-cause higher intensities allow, in general, for controlling fects in a plasma (Marklund and Shukla, 2006). Whereas,faster physical processes, which in turn can be exploited the physics of plasma-based laser electron acceleratorsfor generating correspondingly shorter light pulses. is the main subject covered in Esarey et al., 2009 and Not long after the invention of the laser, available in- Malka, 2011, with a special focus on the different phasestensities were already sufficiently high to trigger non- involved (electron injection and trapping, and pulse prop-linear optical effects like second harmonic generation. agation) and on the role of plasma instabilities in theIt is, however, only after the experimental implementa- acceleration process. Finally, in Ruffini et al., 2010 dif-tion of the Chirped Pulse Amplification (CPA) technique ferent processes related to electron-positron (e+ -e− ) pair(Strickland and Mourou, 1985) that it has been possi- production are reviewed with special emphasis on thoseble to reach the intensity threshold of 1014 -1015 W/cm2 occurring in the presence of highly-charged ions and incorresponding to electric field amplitudes of the same astrophysical environments.order as the Coulomb field in atoms. At such inten- In the present article we address physical processessities the interplay between the laser and the atomic that mainly occur at optical laser intensities mostly largerfield significantly alters the electron’s dynamics in atoms than 1021 W/cm2 , i.e., well exceeding the relativisticand molecules and this can be exploited, for example, threshold. After reporting on the latest technologicalfor generating high-frequency radiation in the extreme- progress in optical and x-ray laser technology (Sec. II),ultraviolet (XUV) and soft-x-ray regions (high-order we review some basic results on the classical and quan-harmonic generation or HHG) (Agostini and DiMauro, tum dynamics of an electron in a laser field (Sec. III).2004; Protopapas et al., 1997). HHG as well as atomic Then, we bridge to lower-intensity physics by reviewingprocesses in intense laser fields have been recently re- more recent advances in relativistic ionization and HHGviewed in Fennel et al., 2010; Teubner and Gibbon, 2009; in atomic gases (Sec. IV). The main subject of the re-and Winterfeldt et al., 2008, with specific emphasis on view, i.e., the response of fundamental systems like elec- 3

trons, photons and even the vacuum to ultra-intense radi- (Yanovsky et al., 2008), while no experiments have beenation fields is covered in Secs. V-X. As will be seen, such performed at this intensity yet. This record intensity hashigh laser intensities represent a unique tool to investi- been reached when the HERCULES laser was upgradedgate fundamental processes like multiphoton Compton to become a 300 TW Ti:Sa system, amplified via CPAscattering (Sec. V), to clarify conceptual issues like radi- and capable of a repetition rate of 0.1 Hz. The 4-gratingation reaction in classical and quantum electrodynamics compressor allowed for a pulse duration of about 30 fs(Sec. VI) and to investigate the structure of the quantum and adaptive optics together with a f /1 parabola en-vacuum (Sec. VII). Also, other fundamental quantum- abled to focus the beam down to a diameter of aboutrelativistic phenomena like the transformation of pure 1.3 µm. This experimental achievement on the laser in-light into massive particles as electrons, muons and pi- tensity pushed the capabilities of a multiterawatt laserons (and their corresponding antiparticles) can become almost to the limit.feasible and can even limit the attainability of arbitrar- The 1-PW threshold has been already reached andily high laser intensities (Secs. VIII-X). Finally, we will even exceeded in various laboratories. For example, thealso review recent suggestions on employing novel high- Texas Petawatt Laser (TPL) at the University of Texas atfrequency lasers and laser-accelerated particle beams to Austin (Texas, USA) has exceeded the Petawatt thresh-directly trigger nuclear and high-energy processes (Secs. old thanks to the OPCPA technique, by compressing anXI-XIII). The main conclusions of the article will be pre- energy of 186 J in a pulse lasting only 167 fs (TPL, 2011).sented in Sec. XIV. The TPL has been employed for investigating laser- Units with ~ = c = 1 and the space-time metric plasma interactions at extreme conditions, particularlyη µν = diag(+1, −1, −1, −1) are employed throughout relevant for astrophysics. Also, the two laser systemsthis review. Vulcan (Vulcan, 2011) and Astra Gemini (Astra Gemini, 2011) at the Central Laser Facility (CLF) in the United Kingdom provide powers of the order of 1 PW. TheII. NOVEL RADIATION SOURCES Vulcan facility can deliver an energy of 500 J in a pulse lasting 500 fs. It is a Nd:YAG laser system amplified In this section we review the latest technical and ex- via CPA and can provide intensities up to 1021 W/cm2 .perimental progress in laser technology. We will discuss Whereas, the Astra Gemini laser consists of two inde-optical and x-ray laser systems separately. The latter pendent Ti:Sa laser beams of 0.5 PW each (energy ofare especially useful for e+ -e− pair production, for di- 15 J and a pulse duration of 30 fs), with a maximumrect laser-nucleus interaction, as well as as probes for focused intensity of 1022 W/cm2 . The particular layoutvacuum-polarization effects (see in particular Secs. VII, of the Astra Gemini laser renders this system especiallyVIII and XI). For overviews of feasible accelerators also versatile for unique applications in strong-field physics,of relevance for the present review see, e.g., Esarey et al., where two ultrastrong beams are required. Two laser2009; Malka, 2011; Nakamura et al., 2010; and Wilson, systems are likely to be updated to the Petawatt level2001 and the relevant original literature as quoted in the in Germany. The first one is the Petawatt High-Energyrespective sections. Laser for heavy Ion eXperiments (PHELIX) Nd:YAG laser at the Gesellschaft für Schwerionenforschung (GSI) in Darmstadt, capable now of delivering an energy ofA. Strong optical laser sources 120 J in about 500 fs (PHELIX, 2011). The 1-PW threshold should be reached by increasing the pulse en- As has been mentioned in the Introduction, since the ergy to 500 J. Combined with the highly-charged ioninvention of the CPA technique (Strickland and Mourou, beams at GSI, the PHELIX facility can be attractive for1985) laser peak intensities have been boosted by sev- experimental investigations in strong-field quantum elec-eral orders of magnitude. Another amplification tech- trodynamics (QED). The second system to be updatednique called Optical Parametric Chirped Pulse Ampli- to 1 PW is the Petawatt Optical Laser Amplifier for Ra-fication (OPCPA), based on the nonlinear interaction diation Intensive Experiments (POLARIS) laser in Jenaamong laser beams in crystals, was suggested almost at (Hein et al., 2010). At the moment, a power of aboutthe same time as the CPA and proved to be promising as 100 TW (energy of 10 J for a pulse duration of 100 fs)well (Piskarskas et al., 1986). As a result of the increase has been reached and the goal of 1 PW power shouldin available laser intensities, exciting perspectives have be achieved by compressing 120 J in about 120 fs. Thebeen envisaged in different fields spanning from atomic to Scottish Centre for the Application of Plasma-based Ac-plasma and even nuclear and high-energy physics (Feder, celerators (SCAPA) research center is one of the main2010; Gerstner, 2007; Mourou, 2010; Tajima et al., 2010). initiatives within the Scottish Universities Physics Al- The group of G. Mourou at the University of Michi- liance (SUPA) project dedicated to the high-power lasergan (Michigan, USA) holds the record so far for the interaction with plasmas. A laser system will be devel-highest laser intensity ever achieved of 2 × 1022 W/cm2 oped, which will generate pulses of 5-7 J energy and of 4

25-30 fs duration at a repetition rate of 5 Hz, correspond- can laser has already started (Vulcan 10PW, 2011). Theing to a peak power of 200-250 TW, with potential for new laser will provide beams with an energy of 300 Jfuture upgrades to the petawatt level (SCAPA, 2012). in only 30 fs via the OPCPA. A 10 PW laser system is The 1-PW threshold has been also exceeded in Ti:Sa in principle capable of unprecedented intensities largerlaser systems like those described in Sung et al., 2010 than 1023 W/cm2 if the beam is focused to about 1 µm.(energy of 34 J for a pulse duration of 30 fs) and in The front-end stage of the Vulcan 10 PW is already com-Wang et al., 2011 (energy of 32.3 J for a pulse duration pleted and it delivers pulses with about 1 J of energy at aof 27.9 fs) and constructed at the Advanced Photonics central wavelength of 0.9 µm, with sufficient bandwidthResearch Institute (APRI) at Gwangju (Republic of Ko- to support a pulse with duration less than 30 fs.rea) and at the Beijing National Laboratory for Con- Another 10 PW laser project is the ILE APOLLONdensed Matter Physics in Beijing (China), respectively. to be realized at the Institut de Lumiére Extreme (ILE)The BELLA (Berkeley Lab Laser Accelerator) is a Ti:Sa in France (Chambaret et al., 2009). The laser pulseslaser system under construction at the Lawrence Berke- are expected to deliver an energy of 150 J in 15 fs atley National Laboratory (LBNL) at Berkeley (California, the last stage of amplification after the front end (en-USA) which will also reach the 1-PW threshold by com- ergy of 100 mJ in less than 10 fs), with a repetition ratepressing 40 J in 40 fs at a repetition rate of 1 Hz (BELLA, of one shot per minute. Laser intensities of the order2012). of 1024 W/cm2 are envisaged at the ILE APOLLON sys- All the above systems require laser energies larger tem, entering the so-called ultrarelativistic regime, wherethan 10 J and this limits the repetition rate of exist- also ions (rest energy of the order of 1 GeV) become rel-ing petawatt lasers in the best situation to about 0.1 Hz ativistic within one laser period of such an intense laser(Sung et al., 2010). The Ti:Sa Petawatt Field Synthe- field.sizer (PFS) system under development in Garching (Ger- We mention here also the PEtawatt pARametric Lasermany) aims to be the first high-repetition rate petawatt (PEARL-10) project at the Institute of Applied Physicslaser system with an envisaged repetition rate of 10 Hz of the Russian Academy of Sciences in Nizhny Novgorod(PFS, 2011). By adopting the OPCPA technique the (Russia), which is an upgrade of the present 0.56-PWPFS should reach the petawatt level by compressing an laser employing the OPCPA technique, to 10 PW (200 Jenergy of about 5 J in 5 fs (Major et al., 2010). For of energy compressed in 20 fs) (Korzhimanov et al.,a recent review on petawatt-class laser systems, see Ko- 2011).rzhimanov et al., 2011. Finally, we also want to mention other high-powerlasers, mainly devoted to fast ignition and characterized 2. Multi-Petawatt and Exawatt optical laser systemsby relatively long pulses of the order of 1 ps-1 ns. Amongothers we mention the OMEGA EP system at Rochester The Extreme Light Infrastructure (ELI) (ELI, 2011)(New York, USA) (energy of 1 kJ for a pulse duration (see Fig. 1), the Exawatt Center for Extreme Lightof 1 ps) (OMEGA EP, 2011) and the National Igni- Studies (XCELS) (XCELS, 2012) and the High Powertion Facility (NIF) at the Lawrence Livermore National laser Energy Research (HiPER) facility at the CLF inLaboratory (LLNL) at Livermore (California, USA) (en- the United Kingdom (HiPER, 2011) are envisaged laserergy of 2 MJ distributed in 192 beams with a pulse du- systems with a power exceeding the 100 PW level.ration of about 3-10 ns) (NIF, 2011). Another high- ELI is a large-scale laser facility consisting of four “pil-power laser facility is the PETawatt Aquitaine Laser lars” (see Fig. 1): one devoted to nuclear physics, one(PETAL) in Le Barp close to Bordeaux (France), which to attosecond physics, one to secondary beams (photonis a multi-petawatt laser, generating pulses with energy beams, ultrarelativistic electron and ion beams) and oneup to 3.5 kJ and with a duration of 0.5 to 5 ps (PETAL, to high-intensity physics. This last one is of relevance2011). The PETAL facility is planned to be coupled to here and it is supposed to comprise ten beams each withthe Laser MégaJoule (LMJ) under construction in Bor- a power of 10-20 PW that, when combined in phase,deaux (France). In the LMJ a total energy of 1.8 MJ is should deliver a single beam of about 100-200 PW at adistributed in a series of 240 laser beamlines, collected repetition rate of one shot per minute. The relativelyinto eight groups of 30 beams with a pulse duration of high repetition rate is obtained by compressing in each0.2-25 ns (LMJ, 2011). beam alone 0.3-0.4 kJ of energy in a pulse of 15 fs. We mention that one of the aims of the ILE APOLLON sys- tem is to provide a prototype of the 10-20 PW beams,1. Next-generation 10 PW optical laser systems that will be then employed at ELI. In the high-field pil- lar of ELI ultrahigh intensities exceeding 1025 W/cm2 The possibility of building a 10 PW laser system is are envisaged, which are well above the ultrarelativisticunder consideration in various laboratories. At the CLF regime. At such intensities, it will be possible to test dif-in the United Kingdom the 10 PW upgrade of the Vul- ferent aspects of fundamental physics for the first time. 5

1035

[photons/(s mrad2 mm2 0.1% bandwidth]

1034 European XFEL 1033

Peak brilliance 1032 LCLS

1031

1030 FLASH 29 10

1028FIG. 1 (Color online) Summary of the four pillars of ELI. A 1 10 102 103 104power value of 10(×2) PW indicates the availability of two Photon energy [eV]laser systems each with 10 PW power. Reprinted with per-mission from Feder, 2010. Copyright 2010, American Instituteof Physics. FIG. 2 (Color online) Comparison among the peak brilliances of the three facilities FLASH, LCLS and European XFEL as a function of the laser photon energy. An envisaged peak brilliance of 5 × 1033 photons/(s mrad2 mm2 0.1% bandwidth) at a photon energy of 12.4 keV for the SACLA facility is reported in European XFEL, 2011. See also European XFEL, The XCELS infrastructure is planned to be built in 2011.Nizhny Novgorod (Russia) and it will consist of 12 beamseach with energy of 300-400 J and with duration of B. Brilliant x-ray laser sources20-30 fs. The pulse resulting from the superposition ofthese beams is expected to have a power of 200 PW, apulse duration of about 25 fs, and divergence less than Strong optical laser systems are sources of coherent3 diffraction limits (at a central wavelength of 0.91 µm). radiation at wavelengths of the order of 1 µm, corre-Apart from aiming to overcome the 100 PW threshold, sponding to photon energies of the order of 1 eV. Con-the main priorities of XCELS are the creation of sources siderable efforts have been devoted in the past few yearsof attosecond and subattosecond, X-ray and γ-ray pulses, to develop coherent radiation sources at photon energiesthe development of laser based electron and ion acceler- larger than 100 eV. The discovery of the Self-Amplifiedators with electron and ion energies exceeding 100 GeV Spontaneous Emission (SASE) regime (Bonifacio et al.,and up to 10 GeV, respectively, the realization in labo- 1984) has opened the possibility of employing Free Elec-ratory of astrophysical and early-cosmological conditions tron Lasers (FELs) to generate coherent light at suchand the investigation of the structure of the quantum short wavelengths. In a FEL relativistic bunches of elec-vacuum. trons pass through a spatially-periodic magnetic field (undulator) and emit high-energy photons. In the SASE regime the interaction of the electron bunch with its own The main goal of the other large-scale facility HiPER electromagnetic field “structures” the bunch itself intois the first demonstration of laser-driven fusion, or fast slices (microbunches) each one emitting coherently evenignition. To this end HiPER will deliver: 1) an energy of at wavelengths below 1 nm (FELs at such small wave-about 200 kJ distributed in 40 beams with a pulse dura- lengths are dubbed X-ray Free Electron Lasers (XFELs)).tion of several nanoseconds and a photon energy of 3 eV The Free-Electron Laser in Hamburg (FLASH) facil-in the compression side; 2) an energy of about 70 kJ dis- ity at the Deutsches Elektronen-SYnchrotron (DESY)tributed in 24 beams with a pulse duration of 15 ps and in Hamburg (Germany) (FLASH, 2011) is one of thea photon energy of 2 eV in the ignition side. Employing most brilliant operating FEL’s. It delivers short pulsesHiPER for high-intensity physics would imply a feasible (duration of about 10-100 fs) of coherent radiation inreconfiguration of the ignition side to deliver 10 kJ in the extreme ultraviolet-soft x-ray regime (fundamentalonly 10 fs via the OPCPA technique. This would render wavelength from 60 nm down to 6.5 nm correspondingHiPER a laser facility with Exawatt (1018 W) power and to photon energies from 21 to 190 eV) at a repetitionwith a potential intensity of 1026 W/cm2 . rate of 100 kHz. The intense electron beams available at FLASH (total charge of 0.5-1 nC at an energy of 1 GeV) Finally, we briefly mention the GEKKO EXA facility allow for peak brilliance of the photon beam of aboutconceptually under design in Osaka (Japan) (GEKKO 1029 -1030 photons/(s mrad2 mm2 0.1% bandwidth) (seeEXA, 2011). This facility is expected to deliver pulses Fig. 2), exceeding the peak brilliance of conventionalof 2 kJ energy and of 10 fs duration corresponding to synchrotron light sources by several orders of magnitude.200 PW and with an intensity up to 1025 W/cm2 . The Linac Coherent Light Source (LCLS) at Stanford 6

(California, USA) uses the electron beams generated by III. FREE ELECTRON DYNAMICS IN A LASER FIELDthe Stanford Linear ACcelerator (SLAC) at the NationalAccelerator Laboratory to generate flashes of coherentx-ray radiation of unprecedented brilliance (LCLS, 2011) In this Section we review, for the benefit of the reader,(see also Emma et al., 2010). Since the electron beam en- some important basic results on the dynamics of a freeergy can be varied from 4.5 GeV to 14.4 GeV, accordingly electron in a laser field (see also the review Eberly, 1969)the wavelength of LCLS can be tuned from 1.5 nm to and link them to recent investigations on the subject.0.15 nm (corresponding to photon energies from 0.8 keV Results in the realm of classical and quantum electro-to 8 keV). The peak brilliance of the LCLS is of the or- dynamics are considered separately. Radiation-reactionder of 1032 -1033 photons/(s mrad2 mm2 0.1% bandwidth) and electron self-interaction effects are not included here(see Fig. 2), the pulse duration is typically of about and their discussion is developed in Sec. VI.40-80 fs up to 500 fs, which can be decreased to 10 fs inthe low-charge electron beam mode, and the repetitionrate is 120 Hz. Another XFEL in operation is the SPring-8 AngstromCompact free electron LAser (SACLA) at the RIKENHarima Institute in Japan (SACLA, 2011). The electronaccelerator, based on a conducting C-band high-gradientradiofrequency acceleration system, and the short-period A. Classical dynamicsundulator allow for a relatively compact facility of around700 m in length (compared, for example, with the about2 km of the LCLS). SACLA employs the 8 GeV electron The motion of a charged particle in a laser field is usu-beam of the Super Photon Ring - 8 GeV (SPring-8) ac- ally associated to an oscillation along the laser polariza-celerator (SPring8, 2011) and has generated x-ray beams tion direction. This is pertinent to the non-relativisticwith 0.08 nm wavelength (corresponding to a photon en- regime, while the charge dynamics in the relativistic do-ergy of 15.5 keV) at a repetition rate of 60 Hz (see also main is enriched by new features like the drift along thethe caption of Fig. 2). laser propagation direction and other non-dipole effects (like the well-known figure-8 trajectory), as well as by The European XFEL is under development at DESY in the sharpening of the trajectory at those instants whereHamburg (Germany) (European XFEL, 2011). It is ex- the velocity along the polarization direction reverses. Aspected to deliver x-ray pulses with a peak brilliance up to a consequence, laser-driven relativistic free electrons alsoabout 5 × 1033 photons/(s mrad2 mm2 0.1% bandwidth) emit high harmonics of the laser frequency (see Sec. V).at the unprecedented repetition rate of 27 kHz. The elec-tron accelerator provides an electron beam with maximal The classical motion of an electron (electric chargeenergy of 17.5 GeV able to generate laser pulses with a e < 0 and mass m) in an arbitrary external electromag-central wavelength of 0.05 nm, which corresponds to a netic field F µν (x) is determined by the Lorentz equationphoton energy of 24.8 keV and with a pulse duration of mduµ /ds = eF µν uν , where uµ = dxµ /ds is the elec-100 fs. Moreover, the European XFEL will be a versatile tron four-velocity and s its proper time (Landau andmachine consisting of three photon beamlines: the SASE- Lifshitz, 1975). If the external field is a plane wave,1 and SASE-2, with linearly polarized photons with en- the field tensor F µν (x) depends only on the dimensionalergy in the range 3.1-24.8 keV and the SASE-3, with phase φ = (n0 x), where nµ0 = (1, n0 ), with n0 being thelinearly or circularly polarized photons of energy in the unit vector along the propagation direction of the wave.range 0.26-3.1 keV. In this case, for an arbitrary four-vector v µ = (v 0 , v) We also mention that coherent attosecond pulses of it is convenient to introduce the notation vk = n0 · v,XUV radiation (photon energy of the order of 100 v⊥ = v − vk n0 and v− = (n0 v) = v 0 − vk . The four-eV) have been generated employing HHG in a gaseous vector potential of the wave can be chosen in the Lorentzmedium (Agostini and DiMauro, 2004). This technique gauge as Aµ (φ) = (0, A(φ)), with A− (φ) = −Ak (φ) = 0.allows for the production of beams with central photon We indicate as pµ = (ε, p) = muµ the (kinetic) four-energy up to several keVs (Popmintchev et al., 2009; San- momentum of the electron. Since a plane-wave field de-sone et al., 2006), though with intensities several orders pends only on φ, the canonical momenta p⊥ (φ) + eA(φ)smaller than XFELs. Less stable sources of coherent soft and p− (φ) are conserved as they are the conjugated mo-x-rays are the so-called x-ray lasers, which are based on menta to the cyclic coordinates x⊥ and t + xk , respec-the amplification of spontaneous emission by multiply tively. For pµ (φ0 ) = pµ0 = (ε0 , p0 ) = mγ0 (1, β0 ) beingionized atoms in dense plasmas created by intense laser the initial condition for the electron’s four-momentum atpulses (Suckewer and Jaegle, 2009; Wang et al., 2008; a given phase φ0 , the above-mentioned conservation lawsZeitoun et al., 2004). already allow to write the electron’s four-momentum at 7

an arbitrary phase φ as (Landau and Lifshitz, 1975) has been switched off (A(∞) = 0) has the components p⊥ (∞) = eA(φ0 ) and pk (∞) = e2 A2 (φ0 )/2m. p0,⊥ · [A(φ) − A(φ0 )] Realistic laser pulses, as those produced in laborato- ε(φ) =ε0 − e p0,− ries, have a more complicated structure than a plane (1) e2 [A(φ) − A(φ0 )]2 wave, essentially because they are spatially focused on + , the transverse planes and the area of the focusing spot 2 p0,− changes along the laser’s propagation axis. Generally p⊥ (φ) =p0,⊥ − e[A(φ) − A(φ0 )], (2) speaking, if the radius of the minimal focusing area (spot p0,⊥ · [A(φ) − A(φ0 )] radius) is much larger than the central wavelength of the pk (φ) =p0,k − e p0,− laser pulse, then the pulse can be reasonably approxi- (3) mated by a plane wave. A Gaussian beam in the paraxial e2 [A(φ) − A(φ0 )]2 + , approximation offers a more accurate analytical descrip- 2 p0,− tion of a realistic laser pulse, which shows a Gaussian pro-where the on-shell condition ε(φ) + pk (φ) = [p2⊥ (φ) + file in the transverse planes (Salamin and Keitel, 2002).m2 ]/p0,− was employed. For the paradigmatic case of The dynamics of an electron in such a field cannot be de-a linearly polarized monochromatic plane wave, it is rived analytically and a numerical solution of the LorentzAµ (φ) = Aµ0 cos(ω0 φ), with Aµ0 = (0, E0 u/ω0 ), where E0 equation is required (Salamin et al., 2002).is the electric field amplitude, ω0 the angular frequencyand u the polarization direction (perpendicular to n0 ). B. Quantum dynamics The above analytical solution indicates that even if anelectron is initially at rest, it becomes relativistic within In the realm of relativistic quantum mechanics, i.e.,one laser period T0 = 2π/ω0 if the parameter when e+ -e− pair production is negligible (see also Sec. p VIII) and the single-particle quantum theory is appli- |e| −A20 |e|E0 ξ0 = = (4) cable, the dynamics of an electron in an external elec- m mω0 tromagnetic field with four-vector potential Aµ (x) is de-is of the order of or larger than unity. In the rela- scribed by the Dirac equationtivistic regime the magnetic component of the Lorentzforce, which depends on the electron’s velocity, becomes {γ µ [i∂µ − eAµ (x)] − m}Ψ = 0, (5)comparable to the electric one and the electron’s dy- where γ µ are the Dirac matrices and where Ψ(x) is thenamics becomes highly nonlinear in the laser-field am- four-component electron bi-spinor (Berestetskii et al.,plitude. Thus, the parameter ξ0 is known as classical 1982). Analogously to the classical case, if the exter-nonlinearity parameter. An heuristic interpretation of nal field is a plane wave, the Dirac equation can bethe parameter ξ0 is as the work performed by the laser solved exactly. If pµ0 = (ε0 , p0 ) and σ0 /2 = ±1/2 arefield on the electron in one laser wavelength λ0 = T0 in the electron’s four-momentum and spin at φ → −∞ andunits of the electron mass, which clearly explains why if Aµ (−∞) = 0, the positive-energy (ε0 > 0) solutionrelativistic effects become important at ξ0 & 1. Alter- Ψp0 ,σ0 (x) of Eq. (5) reads (Berestetskii et al., 1982;natively, Eqs. (1)-(3) indicate that the figure-8 trajec- Volkov, 1935)tory has a longitudinal (transverse) extension of the or-der of λ0 ξ02 (λ0 ξ0 ), implying that the electron trajectory e

up ,σdeviates from the unidirectional oscillating one and be- Ψp0 ,σ0 (x) = 1 + n̂0 Â(φ) √ 0 0 eiSp0 , (6) 2p0,− 2V ε0comes nonlinear in the field p amplitude at ξ0 & 1. Notethatp numerically ξ0 = 6.0 I0 [1020 W/cm2 ]λ0 [µm] = where in general v̂ = γ µ vµ for a generic four-vector v µ ,7.5 I0 [10 W/cm ]/ω0 [eV], where I0 = E02 /4π is the 20 2 where up0 ,σ0 is a positive-energy free bi-spinor (Berestet-wave’s peak intensity, and that ξ0 is gauge- and Lorentz- skii et al., 1982), V is the quantization volume, and whereinvariant: the gauge invariance has to be intended with φ e(p0 A(φ′ )) e2 A2 (φ′ ) Z respect to gauge transformations which do not alter the Sp0 (x) = −(p0 x) − dφ′ −dependence of the four-vector potential on φ (see Heinzl −∞ p0,− 2p0,−and Ilderton, 2009 for a thorough analysis of this issue). (7)The solution in Eqs. (1)-(3) also indicates that in the is the classical action of an electron in the planeultrarelativistic regime at ξ0 ≫ 1, the electron acquires wave (Landau and Lifshitz, 1975). The above electrona drift momentum along the propagation direction of the states are known as positive-energy Volkov states. Thelaser field which is proportional to ξ02 , in contrast to the negative-energy states Ψ−p0 ,−σ0 (x) can be formally ob-transverse momentum which is proportional to ξ0 . In tained by the replacements pµ0 → −pµ0 and σ0 → −σ0 inthe case of an electron initially at rest, for example, the Eq. (6) except for the energy in the square root (the re-momentum p(∞) of the electron after the laser pulse sulting bi-spinor u−p0 ,−σ0 is the corresponding negative- 8

−(q0 x) + “oscillating terms”, with (Ritus, 1985)

m2 ξ02 µ q0µ = pµ0 + n . (8) 4p0,− 0

The four-vector q0µ plays the role of an “effective” four-

momentum of the electron in the laser field and it is indi- cated as electronp “quasimomentum”. p The corresponding electron “mass” q02 = m∗ = m 1 + ξ02 /2 is known as electron’s dressed mass. The results for the quasimo-FIG. 3 (Color) Free wave packet evolution in a plane wave mentum q0µ and the dressed mass m∗ in the case of aﬁeld. The solid gray line indicates the center of mass tra- circularly polarized laser field with the same amplitudejectory, coinciding essentially with the classical trajectory, and frequency is obtained from the above ones with theand the laser pulse travels from left to right. The blue re- replacement ξ02 → 2ξ02 . The quasimomentum coincidesgions indicate the copropagating self-adaptive numerical grid. classically with the average momentum of the electronTime and space coordinates are given in “atomic units”, with1 a.u. = 24 as and 1 a.u. = 0.05 nm, respectively. From in the plane wave. Correspondingly, the mass dressingBauke and Keitel, 2011. depends only on the classical nonlinearity parameter ξ0 and it is an effect of the quivering motion of the electron in the monochromatic wave (see also the recent review Ehlotzky et al., 2009). As we will see in Sec. V.A, itenergy free bi-spinor 1 (Berestetskii et al., 1982)). Al- is important that conservation laws in QED processesthough it has been shown long ago that positive- and in the presence of a monochromatic plane-wave field in-negative-energy Volkov states form a complete set of or- volve the quasimomentum q0µ for the incoming electronsthogonal states on the hypersurfaces φ = const (Ritus, rather than the four-momentum pµ0 . The question of the1985), the corresponding property on the hypersurfaces electron dressed mass in pulsed laser fields has been in-t = const is not straightforward and it has been proved vestigated in Heinzl et al., 2010a and Mackenroth andonly recently (see Ritus, 1985 and Zakowicz, 2005, and Di Piazza, 2011.Boca and Florescu, 2010 for a proof of the orthogonal- In the realm of QED the parameter ξ0 can also beity and of the completeness of the Volkov states, respec- heuristically interpreted as the work performed by thetively). laser field on the electron in the typical QED length Since the Volkov states form a basis of the space of λC = 1/m ≈ 3.9 × 10−11 cm (Compton wavelength) inthe solutions of Dirac equation in a plane wave, they can units of the laser photon energy ω0 (see Eq. (4)). Thisbe employed to build electron wave packets and study qualitatively explains why multiphoton effects in a lasertheir evolution. A pedagogical example of laser-induced field become important at ξ0 & 1, such that the laser fieldDirac dynamics is displayed in Fig. 3 for a plane wave has to be taken into account exactly in the calculationswith peak intensity of 6.3 × 1023 W/cm2 and central (Ritus, 1985). In the framework of QED this is achievedwavelength of 2 nm. The figure shows the drift of the by working in the so-called Furry picture (Furry, 1951),wave packet in the propagation direction of the wave, its where the e+ -e− field Ψ(x) is quantized in the presencespreading and its shearing due to non-dipole effects. In of the plane-wave field. This amounts essentially in em-Fillion-Gourdeau et al., 2012 an alternative method of ploying the Volkov (dressed) states and the correspond-solving the time-dependent Dirac equation in coordinate ing Volkov (dressed) propagators (Ritus, 1985) insteadspace is presented, which explicitly avoids the fermion of free particle states and free propagators to computedoubling, i.e., the appearance of unphysical modes when the amplitudes of QED processes. In the Furry picturethe Dirac equation is discretized. the effects of the plane wave are accounted for exactly As in the classical case, we shortly mention here the and only the interaction between the e+ -e− field Ψ(x)paradigmatic case of a monochromatic, linearly polar- and the radiation field F µν (x) ≡ ∂ µ Aν (x) − ∂ ν Aµ (x)ized plane-wave field Aµ (φ) = Aµ0 cos(ω0 φ). In this case is accounted for by means of perturbation theory. Thethe action Sp0 (x) can be written in the form Sp0 (x) = complete evolution of the system “e+ -e− field+radiation field” is obtained by means of the S-matrix Z 4 µ S = T exp −ie d xΨ̄γ ΨAµ , (9)1 We point out that the discussed Volkov states Ψ±p0 ,±σ0 (x) are the so-called Volkov in-states, as they transform into free-states in the limit t → −∞ (Fradkin et al., 1991). Volkov out-states, where T is the time-ordering operator and Ψ̄(x) = which transform into free-states in the limit t → ∞, can be Ψ† (x)γ 0 . For an initial state containing only a single elec- derived analogously and differ from the Volkov in-states only by tron with four-momentum pµ0 , the quantitative descrip- an inconsequential constant phase factor (recall that A(∞) = 0). tion of the interaction between the electron, the laser field 9

and the radiation field involves in particular the gauge- field F0µν , with the replacement F0µν → F µν (x) (Ritus,and Lorentz-invariant quantum parameter 1985). For a monochromatic plane wave with angular fre- p quency ω0 this occurs if ξ0 ≫ 1. As will be seen in Sec. |e| −(F0,µν pν0 )2 p0,− E0 V.A, this condition corresponds, e.g., to the formation χ0 = 3 = , (10) m m Fcr time of multiphoton Compton scattering (∼ m/|e|E0 )where Fcr = m2 /|e| = 1.3 × 1016 V/cm = 4.4 × 1014 G being much shorter than the laser period T0 .being the critical electromagnetic field of QED (Ritus, As has been mentioned, the S-matrix in Eq. (9)1985). The definition of Fcr indicates that a constant describes all possible electrodynamical processes amongand uniform electric field with strength of the order of electrons, positrons and photons. The above consid-Fcr , provides an e+ -e− pair with an energy of the or- erations can be easily adapted for discussing processesder of its rest energy 2m in a distance of the order of involving an initial positron. Whereas, the probabilitythe Compton wavelength λC , implying the instability dPγ /dV dt of a quantum process in a plane-wave field in-of the vacuum under e+ -e− pair creation in the pres- volving an incoming photon, as, e.g., multiphoton e+ -e−ence of such a strong field (Heisenberg and Euler, 1936; pair production, depends on the parameters ξ0 andSauter, 1931; Schwinger, 1951). In Eq. (10) we con- psidered the case of a linearly polarized plane wave of |e| −(F0,µν k ν )2 k− E0 κ0 = = , (13)the form Aµ (φ) = Aµ0 ψ(φ), with ψ(φ) being an arbi- m3 m Fcrtrary function with max |dψ(φ)/dφ| = 1 and we intro-duced the tensor amplitude F0µν = k0µ Aν0 − k0ν Aµ0 , with where k µ = (ω, k) is the four-momentum of the incomingk0µ = ω0 nµ0 . For an ultrarelativistic electron initially photon (see Ritus, 1985 and Secs. VII-IX). For a pho-counterpropagating withprespect to the plane wave it is ton counterpropagating with p respect to the plane waveχ0 = 5.9 × 10−2 ε0 [GeV] I0 [1020 W/cm2 ]. The parame- it is κ0 = 5.9 × 10−2 ω[GeV] I0 [1020 W/cm2 ]. In theter χ0 can be interpreted as the amplitude of the electric case of multiphoton e+ -e− pair production, the param-field of the plane wave in the initial rest-frame of the elec- eter κ0 can be interpreted as the amplitude of the elec-tron in units of the critical field of QED and it controls tric field of the plane wave in units of the critical fieldthe magnitude of pure quantum effects like the photon Fcr in the center-of-mass system of the created electronrecoil in multiphoton Compton scattering and spin ef- and positron (Ritus, 1985). The above remarks on pro-fects. This is why it is known as “nonlinear quantum cesses occurring in a constant or slowly-varying back-parameter”. ground field F µν (x) and involving an incoming electron, Since the probability dPe /dV dt per unit volume and also apply to the case of an p incoming photon once oneunit time of a quantum process is a gauge- and Lorentz replaces χ(x) with κ(x) = |e| |(Fµν (x)k ν )2 |/m3 .invariant quantity, for those processes in a plane-wavefield involving an incoming electron, as, e.g., multiphotonCompton scattering, it can depend only on the two pa-rameters ξ0 and χ0 (Ritus, 1985). For an electromagnetic IV. RELATIVISTIC ATOMIC DYNAMICS IN STRONGfield F µν (x) = (E(x), B(x)) either constant or slowly- LASER FIELDSvarying, the quantity dPe /dV dt, calculated in the lat-ter case in the leading order with respect to the fields’ When super-intense infrared laser pulses, as those de-derivatives, can in principle also depend on the two field scribed in Sec. II.A, impinge on an atom, the latter isinvariants immediately partly or fully ionized (Becker et al., 2002; Keitel, 2001; Protopapas et al., 1997). The ejected elec- 1 µν 1 F (x) = F (x)Fµν (x) = − [E 2 (x) − B 2 (x)], (11) trons experience the typical “zig-zag” motion of a free 4 2 electron in both laser polarization and propagation direc- 1 µν G (x) = F (x)F̃µν (x) = −E(x) · B(x) (12) tions (see Eqs. (2)-(3) and Fig. 3) and will not, in gen- 4 eral, return to the ionic core. With an enhanced bindingwhich identically vanish for a plane wave. In the sec- force on the remaining electrons, the ionization dynamicsond equation F̃µν (x) = ǫµναβ F αβ (x)/2 is the dual becomes increasingly complex and may experience sub-field of F µν (x) and ǫµναβ is the four-dimensional com- tle relativistic and correlation effects. When the bind-pletely anti-symmetric tensor with ǫ0123 = +1 (since ing force of the ionic core and that of the applied laserG (x) is actually a pseudo-scalar function, the probabil- field eventually become comparable, the electrons mayity dPe /dV dt can only depend on G 2 (x)). Note, how- in special cases return to and interact with the parentever, that p if |F (x)|, |G (x)| ≪ min(1, χ2 (x))Fcr 2 , with ion (rescattering (Corkum, 1993; Kuchiev, 1987; Schafer ν 2 3χ(x) = |e| |(Fµν (x)p0 ) |/m , then the dependence of et al., 1993)). This interaction leads, for example, to thedPe /dV dt on F (x) and G (x) can be neglected. In this ejection of other electrons, to the absorption of energy incase the probability dPe /dV dt essentially coincides with a scattering process or to the emission of high-harmonicthe analogous quantity calculated for a constant crossed photons in case of recombination. 10

2 3N −1 2Eacommon intuitive interpretation of the laser induced tun- ×I κ e−2N Ea /3E0 , E0neling fails in the relativistic regime (Reiss, 2008), a re- (14)vised picture has been proposed in Klaiber et al., 2012.The Strong Field Approximation (SFA) (Faisal, 1973; where α = e2 ≈ 1/137 is the fine-structure constant,Keldysh, 1965; Reiss, 1980), which treats in a universal m1 , . . . , mN are the magnetic quantum numbers of theway both the multiphoton and the tunneling regimes of bound electrons, M = N P j=1 m j , l is the orbital quan-strong-field ionization, has also been extended to the rel- q (N ) (N )ativistic regime (Reiss, 1990a,b). Both the PPT theory tum number of the electrons, κ = 2Ip /mN , Ip = PN (0) (0)and the SFA assume that the direct ionization process oc- j=1 (Ip,j − ∆j ), Ip,j is the jth ionization potential ofcurs as a single-electron phenomenon and thus neglects a parent ion, ∆j is the energy of the core excitation,atomic structure effects. Ea = κ3 Fcr is the atomic field, Z|e| is the charge of When the tunneling process proceeds very fast, multi- the residual ion, I is the adimensional overlap integralelectron correlation effects can occur due to the so-called (see Kornev et√al., 2009 for its precise definition) andshake-up processes. Thus, the detachment of one elec- Cκl ≈ (2/ν)ν / 2πν, with ν = Zα/κ. According to thetron from the atom or ion via tunneling modifies the calculations in Kornev et al., 2009, inelastic and collec-self-consistent potential sensed by the remaining elec- tive tunneling effects contribute significantly to the rel-trons and may result, consequently, in the excitation of ativistic ionization dynamics at intensities larger thanthe atomic core (inelastic tunneling). A strong excitation 1018 W/cm2 , thus changing the ionization probability bymay also trigger the simultaneous escape of several elec- more than one order of magnitude (see Fig. 4).trons from the bound state through the potential barrier Spin effects of bound systems in strong laser fields were(collective tunneling). These effects were known to oc- shown to moderately alter the quantum dynamics andcur also in the nonrelativistic regime (Zon, 1999, 2000) its associated radiation via spin-orbit coupling in highly- 11

tribution for low-energy electrons to electron-correlation

effects. A similar experiment on the energy- and angle- resolved photoionization was later reported for xenon at a laser intensity of 1019 W/cm2 (DiChiara et al., 2010). For energies below 0.5 MeV, the yield and the angular distri- bution were shown not to be described by a one-electron strong-field model, but rather involve most likely mul- tielectron and high-energy atomic excitation processes. A further experiment on relativistic ionization of the methane molecule at I0 ∼ 1018 -1019 W/cm2 (Palaniyap- pan et al., 2008) indicated that molecular mechanisms of ionization play no role, and that C5+ ions are produced at these intensities mostly via the cross-shell rescatteringFIG. 5 (Color) (a) Experimental photoelectron spectra for atomic ionization mechanism. All these experimental re-argon at I0 = 1.2 × 1019 W/cm2 and at an angle of 62◦ from sults still await an accurate theoretical description.the laser propagation direction. Analytical results are shown On the computational side, various numerical meth-for all photoelectrons (continuous line) and for the L-shell ods have been developed to describe the laser-driven rel-(dashed line). The angular distributions are at an electron ativistic quantum dynamics in highly-charged ions. Aenergy of (b) 60 keV, (c) 400 keV, and (d) 770 keV. From Fast-Fourier-Transform split-operator code was imple-DiChiara et al., 2008. mented in Mocken and Keitel, 2008 for solving the Dirac equation in 2+1 dimensions by employing adaptive grid and parallel computing algorithms. Another method hascharged ions already at an intensity of ∼ 1017 W/cm2 been developed in Selstø et al., 2009 to solve the 3D(Hu and Keitel, 1999; Walser et al., 2002). More recently Dirac equation by expanding the angular part of thea nonperturbative relativistic SFA theory has been de- wave-function in spherical harmonics. The latter wasveloped, describing circular dichroism and spin effects in applied to hydrogenlike ions in intense high-frequencythe ionization of helium in an intense circularly polarized laser pulses with emphasis on investigating the role oflaser field (Bhattacharyya et al., 2011). Here, two-photon negative-energy states. In Bauke et al., 2011, the clas-ionization has been studied in the nonrelativistic inten- sical relativistic phase-space averaging method, general-sity range 1013 -1015 W/cm2 with a photon energy of 45 ized to arbitrary central potentials, and the enhancedeV, yielding small relative spin-induced corrections of the time-dependent Dirac and Klein-Gordon numerical treat-order of 10−3 . ments are employed to investigate the relativistic ioniza- A series of experiments has been devoted to the mea- tion of highly-charged hydrogenlike ions in short intensesurement of atomic multi-electron effects in relativisti- laser pulses. For ionization dynamics beyond the tunnel-cally strong laser fields. In DiChiara et al., 2008, the en- ing regime, quantum mechanical and classical methodsergy distribution of the ejected electrons and the angle- give similar results, for laser wavelengths from the near-resolved photoelectron spectra for atomic photoioniza- infrared region to the soft x-ray regime. Furthermoretion of argon at I0 ∼ 1019 W/cm2 have been inves- a useful procedure has been developed, which employstigated experimentally. Here, isolation of the single- the over-the-barrier ionization yields for highly-chargedatom response in the multicharged environment has been ions, to determine the peak laser field strength of shortachieved by measuring photoelectron yields, energies, ultrastrong pulses in the range I0 ∼ 1018 -1026 W/cm2and angular distributions as functions of the sample den- (Hetzheim and Keitel, 2009). In addition, in this arti-sity. Ionization of the entire valence shell along with sev- cle the ionization angle of the ejected electrons is inves-eral inner-shell electrons was shown at I0 ∼ 1017 -1019 tigated by the full quantum mechanical solution of theW/cm2 . A typical spectrum in the case of linear polar- Dirac equation and the laser field strength is shown to beization is displayed in Fig. 5. An extended plateau-like also linked to the electron emission angle. The magneticstructure appears in the spectrum due to the electrons field-induced tilt in the lobes of the angular distributionsoriginating from the L-shell and the longitudinal com- of photoelectrons in laser-induced relativistic ionizationponent of the focused laser field. A surprising feature has also been discussed in Klaiber et al., 2007.is observed in the energy-resolved angular distribution. There are also several new theoretical results for theIn contrast to the nonrelativistic case with increasing ionic quantum dynamics in strong high-frequency laserrescatterings and, thus, angular-distribution widths at fields, in the so-called stabilization regime, where thehigh energies, here azimuthally isotropic angular distri- ionization rate decreases or remains constant also withbutions are observed at low energies (∼ 60 keV in Fig. 5), increasing laser intensity. An unexpected nondipole ef-which become narrower for high-energy photoelectrons. fect has been reported in Førre et al., 2006 via numeri-The authors attribute the anomalous broad angular dis- cally solving the Schrödinger equation for a hydrogenic 12

field may lead to a very broad energy spectrum of emit-

ted recombination photons, with pronounced side wings, and to characteristic modifications of the photon angular distribution. Specific features of nondipole quantum dynamics in strong and ultrashort laser pulses have also been inves- tigated employing the so-called Magnus approximation (Dimitrovski et al., 2009). The dominant nondipole ef- fect is found to be a shift of the entire wave-function to- wards the propagation direction, inducing a substantial population transfer into states with similar geometry. The recent experiment reported in Smeenk et al., 2011 addresses the question of how the photon momenta areFIG. 6 (Color online) Dipole (left) and nondipole (right) shared between the electron and ion during laser-inducedprobability densities of the Kramers-Henneberger wave- multiphoton ionization. Theoretically, this problem re-function in the x-z plane for a x-polarized, 10-cycle sin-like quires a nondipole treatment, even in the nonrelativis-pulse propagating in the positive z direction (upward), withE0 = 1.5 × 1011 V/cm and ω0 = 54 eV. The snapshots are tic case, to take into account explicitly the laser photontaken at t = 0, t = T0 /2, and t = 1.8 T0 from top to bot- momentum. Energy-conservation of ℓ-photon ionizationtom. The length of the horizontal line corresponds to about here means that ℓω0 = Ip + Up + K, where Ip is the ion-50 aB ≈ 2.7 nm, with aB = λC /α ≈ 5.3 × 10−9 cm being ization energy of the atom, Up = e2 E02 /4mω02 is the pon-the Bohr radius . Note that the scale is logarithmic with four deromotive energy and K is the electron’s kinetic energy.contours per decade. From Førre et al., 2006. The experimental results in Smeenk et al., 2011, obtained using laser fields with wavelength of 0.8 µm and 1.4 µm in the intensity range of 1014 -1015 W/cm2 has show that theatom beyond the dipole approximation. For this purpose fraction of the momentum, corresponding to the numberthe Kramers-Henneberger transformation (Henneberger, of observed photons needed to overcome the ionization1968; Kramers, 1956) has been employed, i.e., the trans- energy Ip , is transferred to the created ion rather thanformation to the instantaneous rest-frame of a classical to the photoelectron. The electron carries only the mo-free electron in the laser field, and the terms ∼ ξ02 have mentum corresponding to the kinetic energy K, whilebeen neglected in the Hamiltonian (the value of ξ0 con- the ponderomotive energy and the corresponding por-sidered was approximately 0.14). In Fig. 6, the result- tion of the momentum are transferred back to the lasering angular distribution of the ejected electrons in the field. This experiment shows that the tunneling conceptnondipole regime of stabilization displays a third unex- for the ionization dynamics is only an approximation. Inpected lobe anti-parallel to the laser propagation direc- fact, the quasistatic tunneling provides no mechanism totion, together with the two expected lobes along the laser transfer linear momentum to the ion, a conclusion thatpolarization direction. As a classical explanation, a drift agrees with recent concerns in Reiss, 2008.along the laser propagation direction was identified forthe bound electron wave packet in the nondipole case (seethe middle panel of Fig. 6). Inside the laser field the elec- B. Recollisions and high-order harmonic generationtron has a velocity component along the positive z axisbut this velocity tends to zero at the end of the pulse. Tunneling in the nonrelativistic regime is generally fol-Thus, the electromagnetic forces alone do not change the lowed by recollisions with the parent ion along with var-electron momentum along the propagation direction at ious subsequent effects (Corkum, 1993; Kuchiev, 1987;the end of the pulse. The net effect of the Coulomb forces Schafer et al., 1993). A characteristic feature of strong-on the electron wave packet is consequently a momentum field processes in the relativistic regime is the suppressioncomponent along the negative z axis: the electron, which of recollisions due to the magnetically induced relativisticis most probably situated in the upper hemisphere over drift of the ionized electron in the laser propagation direc-the pulse, undergoes a momentum kick in the negative z tion (see Sec. III.A). Although relativistic effects becomedirection each time it passes close to the nucleus. A sim- significant when the parameter ξ0 exceeds unity, signa-ilar effect has been reported for molecules (Førre et al., tures of the drift in the laser propagation direction can be2007). observed already in the weakly relativistic regime ξ0 . 1. Radiative recombination, being the time-reversed pro- The drift will have a significant impact on the electron’scess of photoionization, of a relativistic electron with a rescattering probability if, at the instant of recollision,highly-charged ion in the presence of a very intense laser the drift distance dk in the laser propagation direction isfield has been considered in Müller et al., 2009. It was larger than the electron’s wave packet size awp,k in thatshown that the strong coupling of the electron to the laser direction (Palaniyappan et al., 2006). The drift distance 13

is approximately given by dk ∼ λ0 ξ02 /2 (see, e.g., Eq.

(3)). Instead, the wave packet size awp,k can be estimatedfrom awp,k ∼ vk ∆t, where vk is a typical electron velocityalong the laser propagation direction and ∆t is the excur-sion time of the electron in the continuum. The velocityvk can be related to the tunneling time τtun via the time-energy uncertainty: mvk2 /2 ∼ 1/τtun. In turn, one can FIG. 7 (Color) The HHG setup with two counterpropagating APTs. After ionization by the laser pulse 1, the ejected elec-estimate the tunneling time τtun as τtun ∼ ltun /vpb , where tron is driven in the same pulse (light blue), propagates freelyltun ∼ Ip /|e|E0 is the tunneling length and vb ∼ 2Ip /m after the pulse 1 has left (gray dashed) and is driven back tothe velocity of the bound electron. In the above esti- the ion by the laser pulse 2 (dark blue). From Kohler et al.,mate, it was assumed that the work carried out by the 2011.laser field along the tunneling length equals Ip . Thus, atthe rescattering moment q ∆t ∼ T0 , the wave packet size pawp,k is of the order of λ0 |e|E0 / m3 Ip and the role of is reduced by the increase of the laser frequency in thethe drift can be characterized by means of the parameter system’s center of mass, an equally strong drift via twor = (dk /awp,k )2 as estimated by constituents with equal mass or via appropriate initial p momenta from antisymmetric orbitals, respectively. 2mIp r∼ ξ03 . (15) On the other hand, the laser field can also be modi- 16ω0 fied to suppress the relativistic drift by employing tightlyThe condition r & 1 determines the parameter region focused laser beams (Lin et al., 2006), two counter-over which the signature of the drift becomes conspicu- propagating laser beams with linear polarization (Kei-ous. tel et al., 1993; Kylstra et al., 2000; Taranukhin, 2000; As an alternative view on the relativistic drift, the ion- Taranukhin and Shubin, 2001, 2002) or equal-handed cir-ized electron here misses the ionic core when it is ionized cular polarization (Milosevic et al., 2004). In the firstwith zero momentum. Nevertheless, the recollision will two cases, the longitudinal component in the tightly fo-occur if the electron is ionized with an appropriate initial cused laser beam may counteract the drift, or the Lorentzmomentum pd (∼ mξ02 /4, see Eq. (3)), opposite to the force may be eliminated in a small area near the antin-laser propagation direction. The probability Pi (pd ) of odes of the resulting standing wave, respectively. In thethis process is exponentially damped, though, due to the third case involving circularly polarized light, the rela-nonzero momentum pd (see, e.g., Salamin et al., 2006): tivistic drift is eliminated because the electron velocity is oriented in the same direction as the magnetic field. 2 (2mIp )3/2 p2d This setup is well suited for imaging attosecond dynam-

Pi (pd ) ∼ exp − 1+ . (16) ics of nuclear processes but not for HHG because of the 3 m|e|E0 4mIp phase-matching problem (Liu et al., 2009). In the weaklyThe drift term pin the exponent proportional to p2d will be relativistic regime the Lorentz force may also be com- 2 pensated by a second weak laser beam polarized alongimportant if 2mIp pd /m|e|E0 & 1, which is equivalentto the condition r & 3. At near-infrared wavelengths the direction of propagation of the strong beam (Chirilă(ω0 ≈ 1 eV) and for the ionization energy Ip = 13.6 eV of et al., 2002). Furthermore, the relativistic drift can beatomic hydrogen, it becomes relevant at laser intensities significantly reduced by means of special tailoring of theI0 approximately above 3 × 1016 W/cm2 . Then, HHG driving laser pulse, which strongly reduces the time whenand other recollision phenomena are suppressed. the electron’s motion is relativistic with respect to a si- The attainability of relativistic recollisions would, how- nusoidal laser pulse (Klaiber et al., 2006, 2007). Twoever, be very attractive for ultrahigh HHG (Kohler et al., consecutive laser pulses (Verschl and Keitel, 2007a) or2012) as well as for the realization of laser controlled a single laser field assisted by a strong magnetic fieldhigh-energy (Hatsagortsyan et al., 2006) and nuclear pro- can also be used to reverse the drift (Verschl and Kei-cesses (Chelkowski et al., 2004; Milosevic et al., 2004). tel, 2007b). In addition two strong Attosecond PulseVarious methods for counteracting the relativistic drift Trains (APTs) (Hatsagortsyan et al., 2008) or an infraredhave been proposed, such as by utilizing highly-charged laser pulse assisted by an APT (Klaiber et al., 2008)ions (Hu and Keitel, 2001; Keitel and Hu, 2002) which have been employed to enhance relativistic recollisions.move relativistically against the laser propagation di- In fact, due to the presence of the APT the ionizationrection (Chirilă et al., 2004; Mocken and Keitel, 2004), can be accomplished by one XUV photon absorption andby employing Positronium (Ps) atoms (Henrich et al., the relatively large energy ωX of the XUV-photon with2004), or through preparing antisymmetric atomic (Fis- ωX = Ip + p2d /2m can compensate the subsequent mo-cher et al., 2007) and molecular (Fischer et al., 2006) or- mentum drift pd ∼ mξ02 /4 in the infrared laser field.bitals. Here the impact of the drift of the ionized electron The main motivation for the realization of relativis- 14

tic recollisions is the extension of HHG towards the hard recent theoretical investigations on this process, we dis-x-ray regime with obvious benefits for time-resolved high- cuss its possible applications for producing high-energyresolution imaging. In the past couple of decades, nonrel- photon beams.ativistic atomic HHG (Corkum, 1993; Lewenstein et al.,1994) has been developed as a reliable source of coher-ent XUV radiation and attosecond pulses (Agostini and A. Fundamental considerationsDiMauro, 2004) opening the door for attosecond time-resolved spectroscopy (Krausz and Ivanov, 2009). Non- When an electron is wiggled by an intense laser wave,relativistic HHG in an atomic gas medium allows already it emits electromagnetic radiation. This process occursto generate coherent x-ray photons up to keV energies with absorption of energy and momentum by the electron(Sansone et al., 2006) and to produce XUV pulses shorter from the laser field and it is named as multiphoton Thom-than 100 as (Goulielmakis et al., 2008). The most favor- son scattering or multiphoton Compton scattering, de-able conversion efficiency for nonrelativistic keV harmon- pending on whether quantum effects, like photon recoil,ics is anticipated with mid-infrared driving laser fields are negligible or not. Multiphoton Thomson and Comp-(Chen et al., 2010; Popmintchev et al., 2009). However, ton scattering in a strong laser field have been studiedprogress in this field has slowed down, especially because theoretically since a long time (see Salamin and Faisal,of the inhibition, alluded to above, of recollisions due 1998; Sarachik and Schappert, 1970; and Sengupta, 1949to optical driving-field intensity above 3 × 1016 W/cm2 . for multiphoton Thomson scattering and Brown and Kib-This indicates the limit on the cut-off frequency ωc of ble, 1964; Goldman, 1964; and Nikishov and Ritus, 1964anonrelativistic HHG to ωc ≈ 3.17Up ∼ 10 keV. for multiphoton Compton scattering). The classical cal- Another factor hindering HHG at high intensities is culation of the emitted spectrum is based on the analyt-the less favorable phase-matching. In strong laser fields, ical solution in Eqs. (1)-(3) of the Lorentz equation inouter-shell electrons are rapidly ionized and produce a a plane wave and the substitution of the correspondinglarge free electron background causing a phase mismatch electron trajectory in the Liénard-Wiechert fields (Jack-between the driving laser wave and the emitted x-rays. son, 1975; Landau and Lifshitz, 1975). Whereas, as weThe feasibility of phase-matched relativistic HHG in a have discussed in Sec. III.B, the quantum calculation ofmacroscopic ensemble was first investigated in Kohler the amplitude of the process is performed in the Furryet al., 2011. Here, the driving fields are two counterprop- picture of QED. As a result, the total emission probabil-agating APTs consisting of 100 as pulses with a peak ity depends only on the two Lorentz- and gauge-invariantintensity of the order of 1019 W/cm2 (see Fig. 7). The parameters ξ0 (see Eq. (4)) and χ0 (see Eq. (10)).electron is driven to the continuum by the laser pulse 1 The parameter ξ0 has already been discussed in Sec.in Fig. 7, followed by the usual relativistic drift. There- III.A. In the contest of multiphoton Compton scatteringafter, the laser pulse 2 overtakes the electron, reverses this parameter controls in particular the effective orderthe drift and imposes the rescattering, yielding a much ℓeff of the emitted harmonics, which, for an ultrarela-higher HHG signal than for a conventional laser field at tivistic electron, can be estimated in the following way.the same cut-off energy. Here phase-matching can be ful- In order to effectively emit a frequency ω ′ , the formationfilled due to an additional intrinsic phase specific to this length lf of the process must not exceed the coherencesetup, depending on the time delay between the pulses length lcoh , because, otherwise, interference effects wouldand on the pulse intensity. The latter, being unique for hinder the emission. Since an electron with p instanta-this laser setup, mainly affects the electron excursion neous velocity β and energy ε = mγ = m/ 1 − β 2 ≫ mtime and varies along the propagation direction. The mainly emits along the direction of β, within a cone withphase-matching is achieved by modifying the laser in- apex angle ϑ ∼ 1/γ ≪ 1, the formation length lf can betensity along the propagation direction and by balanc- estimated from lf ∼ ̺/γ, with ̺ being the instantaneousing the phase slip due to dispersion with the indicated radius of curvature of the electron trajectory (Jackson,intrinsic phase. Note, however, that HHG in the rela- 1975). On the other hand, lcoh = π/ω ′ (1 − β cos ϑ) ∼tivistic regime has been observed experimentally rather γ 2 /ω ′ (Baier et al., 1998; Jackson, 1975). By requir-efficiently in laser plasma interactions (Dromey et al., ing that lf . lcoh , we obtain the following estimate2006). for the largest-emitted frequency (cut-off frequency) ωc′ : ωc′ ∼ γ 3 /̺. Now, in the average rest-frame of the elec- tron, i.e., in the reference frame where the average elec-V. MULTIPHOTON THOMSON AND COMPTON tron velocity along the propagation direction of the laserSCATTERING vanishes (see Eq. (3)), it is γ ⋆ ∼ ξ0 (corresponding to the energy ε⋆ ∼ mξ0 ) and lf⋆ ∼ λ⋆0 /ξ0 , where the upper in- In this section we discuss one of the most fundamental dex ⋆ indicates the variable in this frame. Consequently,processes in QED in a strong laser field: the emission of ωc′ ⋆ ∼ ξ03 ω0⋆ and the effective order of the emitted har-radiation by an accelerated electron. After reporting on monics is ℓeff ∼ ξ03 (note that ℓeff is a Lorentz scalar). As 15

the order of the emitted harmonics corresponds quantum recognized as a possible experimental signature of multi-mechanically to the number of laser photons absorbed by photon Compton scattering (Harvey et al., 2009).the electron during the emission process, the parameter First calculations on multiphoton Thomson and Comp-ξ0 is also said to determine the “multiphoton” character ton scattering have mainly focused on the easiest case of aof the process. monochromatic background plane wave, either with cir- On the other hand, the nonlinear quantum parameter cular or linear polarization. The main results of theseχ0 (see Eq. (10)) in the contest of multiphoton Compton investigations, like the dependence of the emitted fre-scattering controls the importance of quantum effects as quencies on the laser intensities have been recently re-the recoil of the emitted photon. In fact, we can esti- viewed in Ehlotzky et al., 2009. The complete descriptionmate classically the importance of the emitted photon of the multiphoton Compton scattering process with re-recoil from the ratio ωc′ /ε and our considerations above spect to the polarization properties of the incoming andexactly indicate that ωc′ /ε ∼ ξ02 ω0⋆ /m ∼ χ0 . Thus, multi- the outgoing electrons, and of the emitted photon in aphoton Thomson scattering is characterized by the con- monochromatic laser wave has been presented in Ivanovdition χ0 ≪ 1, while multiphoton Compton scattering et al., 2004. Recently significant attention has been de-by χ0 & 1. This result can also be obtained in the case voted to the investigation of multiphoton Thomson andof a monochromatic laser wave starting from the energy- Compton scattering in the presence of short and evenmomentum conservation relation ultrashort plane-wave pulses (we recall that such pulses have still an infinite extension in the directions perpen- q0µ + ℓk0µ = q ′µ + k ′µ (17) dicular to the propagation direction). In Boca and Flo- rescu, 2009 multiphoton Compton scattering has beenin the case in which ℓ laser photons are absorbed in the considered in the presence of a pulsed plane wave. Theprocess (Ritus, 1985). Here q0µ and q ′µ are the quasimo- angular-resolved spectra are practically insensitive to thementa of the initial and final electron (see Eq. (8)) and precise form of the laser pulse for ω0 τ0 ≥ 20, with τ0k ′µ = ω ′ n′µ is the four-momentum of the produced pho- being the pulse duration. The main differences with re-ton (n′2 = 0). From this expression it is easy to obtain spect to the monochromatic case are: 1) a broadening ofthe energy ω ′ of the emitted photon as the lines corresponding to the emitted frequencies; 2) the appearance of sub-peaks, which are due to interference ℓω p ω′ = 0 0,− 2 2 . (18) effects in the emission at the beginning and at the end of m ξ (n′ p0 ) + ℓω0 + 4p0,−0 n′− the laser pulse. On the one hand, the continuous nature of the emission spectrum in a finite pulse in contrast toBy reminding that ℓeff ∼ ξ03 and by estimating the typical the discrete one in the monochromatic case has a clearemission angle of the photon (Mackenroth and Di Piazza, mathematical counterpart. In both cases, in fact, the2011), it is possible to show that ω ′ ∼ χ0 ε0 at ξ0 ≫ 1. As total transverse momenta P⊥ with respect to the laserit has also been throughroughly investigated analytically propagation direction and the total quantity P− are con-and numerically in Boca and Florescu, 2011 and Seipt served in the emission process (see Sec. III). However, inand Kämpfer, 2011a, multiphoton Compton and Thom- the monochromatic case the following additional conser-son spectra coincide in the limit χ0 → 0, although in vation law holds (for a linearly polarized plane wave, seeSeipt and Kämpfer, 2011a differences have been observed Eq. (17))numerically in the detailed structure of the classical and m2 ξ02 m2 ξ02quantum spectra also for χ0 ≪ 1. The most important ε0 + p0,k + + 2ℓω0 = ω ′ + kk′ + ε′ + p′k + , (19)difference between classical and quantum spectra is cer- 2p0,− 2p′−tainly the presence of a sharp cut-off in the latter as so that the resulting four-dimensional energy-momentuman effect of the photon recoil: the energy of the photon conservation law allows only for the emission of the dis-emitted in a plane wave is limited by the initial energy crete frequencies in Eq. (18). On the other hand, theof the electron2 . This does not occur classically, as there appearance of sub-peaks has been in particular investi-the frequency of the emitted radiation does not have the gated in Heinzl et al., 2010c, where it has been foundphysical meaning of photon energy. The dependence of that the number Ns-p of sub-peaks within the first har-the energy cut-off on the laser intensity has been recently monic scales linearly with the pulse duration τ0 and with ξ02 : Ns-p = 0.24 ξ02 τ0 [fs]. In this paper the effects of spa- tial focusing of the driving laser pulse are also discussed. The authors investigate in particular the dependence of2 In the case of a plane-wave background field, this limitation ′ the deflection angle αout undergone by the electron af- rather concerns the quantity k− of the emitted photon, as ′ = p ′ ter colliding head-on with a Gaussian focused beam as a k− 0,− −p− < p0,− . However, for an ultrarelativistic electron with p0,− ≫ mξ0 and initially counterpropagating with respect function of the impact parameter b (see Fig. 8). ′ ≈ 2ω ′ and p′ ≈ 2ε′ (Baier to the laser field, it is p0,− ≈ 2ε0 , k− − By further decreasing the laser pulse duration, it has et al., 1998). been argued that effects of the relative phase between the 16

A more general scaling law has been determined in Seipt

and Kämpfer, 2011b, which relaxes the previous assump- tions on head-on collision and on backscattered radiation employed in Heinzl et al., 2010c. Moreover, in Seipt and Kämpfer, 2011a a simple relation is determined between the classical and the quantum spectral densities. Finally, in Boca and Oprea, 2011 it is found that in the ultra- relativistic case γ0 ≫ 1 the angular distribution of the emitted radiation, integrated with respect to the photon energy, only depends on the ratio ξ0 /γ0 and not on the independent values of ξ0 and γ0 (see also Mackenroth et al., 2010).

FIG. 8 (Color online) Deﬂection αout of an electron initially In the above-mentioned publications the spectral prop-counterpropagating with respect to a laser ﬁeld with an en- erties of the emitted radiation in the classical and quan-ergy of 3 J and a pulse duration of 20 fs, as a function of the tum regimes have been considered. In Kim et al., 2009impact parameter b for diﬀerent laser waist radii w0 . FromHeinzl et al., 2010c. and Zhang et al., 2008, instead, the temporal properties of the emitted radiation in multiphoton Thomson scatter- ing have been investigated. In both papers the feasibility of generating single attosecond pulses is discussed.pulse profile and the carrier wave (the so-called CarrierEnvelope Phase (CEP)) should become visible in multi- Photoemission by a single-electron wave packet viaphoton Thomson and Compton scattering. In Boca and Thomson scattering in a strong laser field has been dis-Florescu, 2009 the case of ultrashort pulses with ω0 τ0 & 4 cussed in Peatross et al., 2008. It was shown that thehas been discussed and effects of the CEP on the har- partial emissions from the individual electron momentummonic yield in specific frequency ranges have been ob- components do not interfere when the driving field is aserved. In Mackenroth et al., 2010 the dependence of the plane wave. In other words, the size of the electron waveangular distribution of the emitted radiation in multi- packet, even when it spreads to the scale of the wave-photon Thomson and Compton scattering on the CEP of length of the driving field, does not affect the Thomsonfew-cycles pulses has been exploited to propose a scheme emission.to measure the CEP of ultrarelativistic laser pulses (in-tensities larger than 1020 W/cm2 ). The method is es- Finally, we shortly mention that multiphoton effectssentially based on the high directionality of the photon in Thomson and Compton scattering have been mea-emission by an ultrarelativistic electron, because the tra- sured in various laboratories. The second-harmonic ra-jectory of the electron, in turn, also depends on the laser’s diation was first observed in the collision of a 1 keV elec-CEP. Accuracies in the measurement of the CEP of the tron beam with a Q-switched Nd:YAG laser, althoughorder of a few degrees are theoretically envisaged. Multi- the laser intensity was such that ξ0 ≈ 0.01 (Englertphoton Compton scattering in one-cycle laser pulses has and Rinehart, 1983), and then in the interaction of abeen considered in Mackenroth and Di Piazza, 2011 and mode-locked Nd:YAG laser (ξ0 = 2) with plasma elec-a substantial broadening of the emission lines with re- trons (Chen et al., 1998). Multiphoton Thomson scat-spect to the monochromatic case has been observed. The tering of laser radiation in the x-ray domain has beenhigh-directionality of radiation emitted via multiphoton reported in Babzien et al., 2006 (see Pogorelsky et al.,Thomson scattering has also been employed as a diag- 2000 for a similar proof-of-principle experiment). Single-nostic tool in Har-Shemesh and Di Piazza, 2012, where shot measurements of the angular distribution of the sec-a new rather precise method has been proposed to mea- ond harmonic (photon energy 6.5 keV) at various lasersure the peak intensity of strong laser fields (intensities polarizations have been carried out by employing a 60between 1020 W/cm2 and 1023 W/cm2 ) from the angular MeV electron beam and a subterawatt CO2 laser beamaperture of the photon spectrum. with ξ0 = 0.35. In the prominent SLAC experiment The study of multiphoton Thomson and Compton (Bula et al., 1996) multiphoton Compton emission wasscattering in short laser pulses has also stimulated the in- detected for the first time. In this experiment an ultra-vestigation of scaling laws for the photon spectral density relativistic electron beam with energy of about 46.6 GeV(Boca and Oprea, 2011; Heinzl et al., 2010c; Seipt and collided with a terawatt Nd:glass laser with an intensityKämpfer, 2011a,b). For example, in Heinzl et al., 2010c a of 1018 W/cm2 (ξ0 ≈ 0.8 and χ0 ≈ 0.3) and four-photonscaling law has been found for backscattered radiation in Compton scattering has been observed indirectly via athe case of head-on laser-electron collisions, which sim- nonlinear energy shift in the spectrum of the outcomingplifies the averaging over the electron-beam phase space. electrons. 17

B. Thomson- and Compton-based sources of high-energy synchronized with the driving laser field. In order tophoton beams further improve the all-optical setup, design parameters for a proof-of-concept experiment have been analyzed in The single-particle theoretical analysis presented above Hartemann et al., 2007. For the calculation of the Comp-indicates that high-energy photons can be emitted via ton scattering parameters, a 3D Compton scattering codemultiphoton Thomson and Compton scattering of an ul- has been used, which was extensively tested for Comp-trarelativistic electron. For example, an electron with ton scattering experiments performed at LLNL (Browninitial energy ε0 ≫ m colliding head-on with an optical et al., 2004; Brown and Hartemann, 2004; Hartemannlaser field (ω0 ≈ 1 eV) of moderate intensity (ξ0 . 1) is et al., 2004, 2005) (see Sun and Wu, 2011 for an alter-barely deflected by the laser field (̺ ∼ λ0 γ0 /ξ0 ≫ λ0 ) native numerical simulation scheme). It is shown thatand potentially emits photons with energies ω[keV] . x-ray fluxes exceeding 1021 s−1 and a peak brightness3.8 × 10−3 ε20 [MeV]. This feature has boosted the idea larger than 1019 photons/(s mrad2 mm2 0.1% bandwidth)of so-called Thomson- and Compton-based sources of can be achieved at photon energies of about 0.5 MeV. Ahigh-energy photons as a valid alternative to conven- few years later the Compton-based photon source Mo-tional synchrotron sources, the main advantages of the noEnergetic Gamma-ray (MEGa-ray) has been designedformer being the compactness, the wide tunability, the at LLNL (Gibson et al., 2010). Production of gamma-shortness of the photon beams in the femtosecond scale rays ranging from 75 keV to 0.9 MeV has been demon-and the potential for high brightness. Unlike the exper- strated with a peak spectral brightness of 1.5 × 1015 pho-iments on multiphoton Thomson and Compton scatter- tons/(s mrad2 mm2 0.1% bandwidth) and with a flux ofing where laser systems with ξ0 & 1 are generally em- 1.6 × 105 photons/shot. An experimental setup for high-ployed, Thomson- and Compton-based photon sources flux gamma-ray generation has been constructed in thepreferably require lasers with ξ0 . 1, such that multipho- Saga Light-Source facility in Tosu (Japan), by collid-ton effects are suppressed and shorter bandwidths of the ing a 1.4 GeV electron beam with a CO2 laser (wave-photon beam are achieved. On the other hand, the elec- length 10.6 µm) (Kaneyasu et al., 2011). A flux of abouttron beam quality is crucial for Thomson- and Compton- 3.2 × 107 photons/s gamma photons with energy largerbased radiation sources. In particular, the brightness of than 0.5 MeV has been obtained.the photon beam scales inversely quadratically with the In the basic setups of Thomson- and Compton-basedelectron beam emittance, and linearly with the electron photon sources the electrons experience the intense laserbunch current density. field for a time-interval much shorter than that needed Proof-of-principle experiments have demonstrated to cross the whole laser beam, the former being of the or-Thomson- and Compton-based photon sources by cross- der of the laser’s Rayleigh length divided by the speed ofing a high-energy laser pulse with a picosecond relativis- light. Thus, the quest for a more intense laser pulse at atic electron beam from a conventional linear electron ac- given power in order to increase the photon yield impliescelerator (Chouffani et al., 2002; Leemans et al., 1996; a tighter focusing and therefore a shorter effective inter-Pogorelsky et al., 2000; Sakai et al., 2003; Schoenlein action time, which in turn causes a broadening of theet al., 1996; Ting et al., 1995, 1996). We also mention photon spectrum. In Debus et al., 2010 the Traveling-the benchmark experiment carried out at LLNL, where Wave Thomson Scattering (TWTS) setup is proposed,photons with an energy of 78 keV have been produced which allows the electrons to stay in the focal regionwith a total flux of 1.3 × 106 photons/shot, by collid- of the laser beam during the whole crossing time (seeing an electron beam with an energy of 57 MeV with a Fig. 9). This is achieved by employing cylindrical op-Ti:Sa laser beam with an intensity of about 1018 W/cm2 tics to focus the laser field only along one direction (red(ξ0 ≈ 0.5) (Gibson et al., 2004). lines in Fig. 9) and, depending on the angle between Another achievement in the development of Thomson- the initial electron velocity and the laser wave-vector, byand Compton-based photon sources has been the exper- tilting the laser pulse front. As a result, an interactionimental realization of a compact all-optical setup, where length ∼ 1 cm-1 m can be achieved and, correspondinglythe electrons are accelerated by an intense laser. In very large photon fluxes (up to 5 × 1010 photons/shotthe first experiment with an all-optical setup (Schwoerer at 20 keV). An alternative way of reaching longer effec-et al., 2006), x-ray photons in the range of 0.4 keV to 2 tive laser-electron interaction times has been proposed inkeV have been generated. In this experiment the elec- Karagodsky et al., 2010, where a planar Bragg structuretron beam was produced by a high-intensity Ti:Sa laser is employed to guide the laser pulse and realize Thom-beam (I0 ≈ 2 × 1019 W/cm2 ) focused into a pulsed he- son/Compton scattering in a waveguide. In this way,lium gas jet. The characteristic feature of the all-optical the yield of x-rays can be enhanced by about two orderssetup is that the electron bunches and, consequently, the of magnitude with respect to the conventional free-spacegenerated x-ray photon beams have an ultrashort dura- Gaussian-beam configuration at given electron beam andtion (∼ 100 fs) and a linear size of the order of 10 µm. injected laser power in both configurations. However,Another advantage is that the electrons can be precisely there are two constraints specific to this setup. On the 18

Normalized on-axis brightness

FIG. 9 (Color) Schematic setup of TWTS with the red linesindicating the laser focal lines. In the notation of Debus et al.,2010 φ is the angle between the initial electrons’ velocity andthe laser’s wave vector and α is the angle between the laserpulse front and the laser propagation direction. Adapted fromDebus et al., 2010. ω ′ /ε0

FIG. 10 (Color online) Normalized on-axis brightness for dif-

one hand, the electron beam has to have a small angular ferent values of the rapidity ρ = cosh−1 γ0 at a plasma tem-spread in order to be injected into the planar Bragg struc- perature of 20 keV. See Hartemann et al., 2008 for the mean- ing of the blue circles at ρ = 6. Adapted from Hartemannture without causing wall damage. On the other hand, et al., 2008.the laser field strength has to be such that ξ0 . 8 × 10−4to avoid surface damage.

Finally, in Hartemann et al., 2008 a setup has been

VI. RADIATION REACTIONproposed to obtain bright GeV gamma-rays via Comp-ton scattering of electrons by a thermonuclear plasma.In fact, a thermonuclear deuterium-tritium plasma pro- The issue of “radiation reaction” (RR) is one of theduces intense blackbody radiation with a temperature oldest and most fundamental problems in electrodynam-∼ 20 keV and a photon density ∼ 1026 cm−3 (Tabak ics. Classically it corresponds to the determination ofet al., 1994). When a thermal photon with energy the equation of motion of a charged particle, an electronω ∼ 1 keV counterpropagates with respect to a GeV for definiteness, in a given electromagnetic field F µν (x).electron (γ0 ∼ 103 ), a Doppler-shifted high-energy pho- In fact, the Lorentz equation mduµ /ds = eF µν uν (seeton ω ′ ∼ γ02 ω ∼ 1 GeV can be emitted on axis, i.e., in Sec. III.A) does not take into account that the electron,the same direction of the incoming electron. Since ω ′ while being accelerated, emits electromagnetic radiationhas to be smaller than the initial electron energy ε0 , a and loses energy and momentum in this way. The firstkinematical photon pileup is induced in the emitted pho- attempt of taking into account the reaction of the radia-ton spectrum at ε0 (Zeldovich and Sunyaev, 1969) (see tion emitted by the electron on the motion of the electronFig. 10). This results in a quasimonochromatic GeV itself (from here comes the expression “radiation reac-gamma-ray beam with a peak brightness & 1030 pho- tion”), was accomplished by H. A. Lorentz in the nonrel-tons/(s mrad2 mm2 0.1% bandwidth), comparable with ativistic regime (Lorentz, 1909). Starting from the knownthat of the FLASH (see Sec. II.B). Larmor formula PL = (2/3)e2 a2 for the power emitted by an electron with instantaneous acceleration a, Lorentz The unique features of Thomson- and Compton-based argued that this energy-loss corresponds to a “damping”photon sources render them a powerful experimental de- force FR = (2/3)e2 da/dt acting on the electron. Thevice. For example, they can be employed for medi- expression of the damping force was generalized to thecal radio-isotope production and photo-fission, and for relativistic case by M. Abraham in the form (Abraham,studying nuclear resonance fluorescence for in situ iso- 1905)tope detection (Albert et al., 2010). In this respect, suchphoton sources will represent the main experimental toolfor nuclear-physics investigations at the Romanian pil- d2 uµ duν duν µ

where the meaning of the symbol m0 for the electron Fcr

λ ≫ αλC , F ≪ , (23)mass will be clarified below. In order to write an “ef- αfective” equation of motion for the electron which in- where Fcr is the critical electromagnetic field of QED (seecludes RR, one “removes” the degrees of freedom of Sec. III.B). This allows for the reduction of order in thethe electromagnetic field (Teitelboim, 1971). This is LAD equation, i.e., for the substitution of the electronachieved in Landau and Lifshitz, 1975 at the level of acceleration in the RR force via the Lorentz force dividedthe Lagrangian of the system electron+electromagnetic by the electron mass. In order to perform the analogousfield and an interesting connection of the RR problem reduction of order in the relativistic case, the conditionswith the derivation of the so-called Darwin Lagrangian (23) have to be fulfilled in the instantaneous rest-frameis indicated. By working at the level of the equations of the electron (Landau and Lifshitz, 1975). The resultof motion (21), one first employs the Green’s function is the so-called Landau-Lifshitz (LL) equationmethod and formally determines the retarded solution µνFT,ret (x) of the (inhomogeneous) Maxwell’s equations: duµ 2 he m =eF µν uν + e2 (∂α F µν )uα uν µνFT,ret µν (x) = F µν (x) + FS,ret (x) (Teitelboim, 1971). Sub- ds 3 m e2 µν e2

µνstitution of FT,ret (x) in the Lorentz equation eliminates α αν − 2 F Fαν u + 2 (F uν )(Fαλ u )u . λ µthe electromagnetic field’s degrees of freedom, but it is m mnot straightforward because FT,ret µν (x) has to be calcu- (24)lated at the electron’s position, where the electron cur- The LL equation is not affected by the shortcomings ofrent diverges. This difficulty is circumvented by model- the LAD equation: for example, it is evident that if theing the electron as a uniformly charged sphere of radius a external field vanishes, so does the electron acceleration.tending to zero. After performing the substitution, and Most importantly, the conditions (23) in the instanta-by neglecting terms which vanish in the limit a → 0, one neous rest-frame of the electron have always to be fulfilledobtains the equation (m0 + δm)duµ /ds = eF µν uν + FRµ in the realm of classical electrodynamics, i.e., if quantumwith δm = (4/3)e2 /a being formally diverging. How- effects are neglected. In order for this to be true, in fact,ever, it is important to note that the only diverging term the two weaker conditions λ ≫ λC and F ≪ Fcr havein the limit a → 0 is proportional to the electron four- to be fulfilled in the instantaneous rest-frame of the elec-acceleration. At this point a sort of “classical renor- tron: the first guarantees that the electron’s wave func-malization principle” is employed, saying that what one tion is well localized and the second ensures that puremeasures experimentally as the physical electron mass m quantum effects, like photon recoil or spin effects areis the overall coefficient of the four-acceleration duµ /ds. negligible (Baier et al., 1998; Berestetskii et al., 1982;Therefore, one sets m = m0 + δm and obtains the so- Ritus, 1985) (see also Sec. III.B). This observation ledcalled Lorentz-Abraham-Dirac (LAD) equation: F. Rohrlich to state recently that the LL equation is the duµ 2 2 d2 uµ

duν duν µ “physically correct” classical relativistic equation of mo- µν m = eF uν + e + u . (22) tion of a charged particle (Rohrlich, 2008). Rohrlich’s ds 3 ds2 ds ds statement is also supported by the findings in Spohn,We point out that renormalization in quantum field the- 2000, where it is shown that the physical solutions ofory is based on the fact that the bare quantities, like the LAD equation, i.e., those which are not runaway-charge and mass, appear in the Lagrangian density of the like, are on the critical manifold of the LAD equationtheory, which is not an observable physical quantity. On itself and are governed there exactly by the LL equa-the other hand, the bare electron mass m0 , which is for- tion. On the other hand, since the LL equation is derivedmally negatively diverging for m to be finite, appears here from the LAD equation, one may still doubt on its rigor-in the system of equations (21), which should “directly” ous validity, due to the application in the latter equationprovide classical physical observables, like the electron of the “suspicious” classical mass-renormalization proce-trajectory. On the other hand, it is also known that the dure. However, this procedure is avoided in Gralla et al.,LAD equation is plagued with physical inconsistencies 2009 by employing a more sophisticated zero-size limit-like, for example, the existence of the so-called “runaway” ing procedure, where also the charge and the mass of the 20

particle are sent to zero but in such a way that their ra- term in the RR force, i.e., the term proportional to thetio remains constant. The authors conclude that at the derivative of the electron acceleration (see Eq. (20)),leading-order level the LL equation represents the self- is thoroughly investigated and in Kazinski and Shipulya,consistent perturbative equation of motion for a charge 2011 the asymptotics of the physical solutions of the LADwithout electric and magnetic moment. The motion of equation at large proper times are obtained. Whereas, ina continuous charge distribution interacting with an ex- Noble et al., 2011 a kinetic theory of RR is proposed,ternal electromagnetic field is also investigated by a self- based on the LAD equation and applicable to study sys-consistent model and at a more formal level in Burton tems of many particles including RR (this last aspectet al., 2007. is also considered in Rohrlich, 2007). Enhancement of From the original derivation of the LL equation from RR effects due to the coherent emission of radiation by athe LAD equation in Landau and Lifshitz, 1975, it is large number of charges is discussed in Smorenburg et al.,expected that the two equations should predict the same 2010. In this respect, we mention here that, in order toelectron trajectory, possibly with differences smaller than investigate strong laser-plasma interactions at intensitiesthe quantum effects. This conclusion has been recently exceeding 1023 W/cm2 , RR effects have been also imple-confirmed by analytical and numerical investigations in mented in Particle-In-Cell (PIC) codes (Tamburini et al.,Hadad et al., 2010 for an external plane-wave field with 2012, 2010; Zhidkov et al., 2002) by modifying the Vlasovlinear and circular polarization and in Bulanov et al., equation for the electron distribution function according2011 for different time-dependent external electromag- to the LL equation. Specifically, in Zhidkov et al., 2002netic field configurations. An effective numerical method it is shown that in the collision of a laser beam withto calculate the trajectory of an electron via the LL equa- intensity I0 = 1023 W/cm2 with an overdense plasmation, which explicitly maintains the relativistic covari- slab, about 35% of the absorbed laser energy is con-ance and the mass-shell condition u2 = 1, has been ad- verted into radiation and that the effect of RR amountsvanced in Harvey et al., 2011a. An alternative numerical to about 20%. One-dimensional (Tamburini et al., 2010)method for determining the dynamics of an electron in- and three-dimensional (Tamburini et al., 2012) PIC sim-cluding RR effects has been proposed in Mao et al., 2010. ulations have shown that RR effects strongly depend on We should emphasize that the LL equation is not the the polarization of the driving field: while for circular po-only equation which has been suggested to overcome the larization they are negligible even at I0 ∼ 1023 W/cm2 ,inconsistencies of the LAD equation. A list of alternative at those intensities they are important for linear polariza-equations can be found in the recent review (Hammond, tion. The simulations also show the beneficial effects of2010) (see also Seto et al., 2011). A phenomenological RR in reducing the energy spread of ion beams generatedequation of motion, including RR and quantum effects via laser-plasma interactions (see also Sec. XII.A). A dif-related to photon recoil, has been suggested in Sokolov ferent beneficial effect of RR on ion acceleration has beenet al., 2010a, 2009 (see also Sec. VI.B). The authors write found in Chen et al., 2011 for the case of a transparentthe differential variation of the electron momentum as plasma: RR strongly suppresses the backward motion ofdue to two contributions: one arising from the external the electrons, cools them down and increases the num-field and one corresponding to the recoil of the emitted ber of ions to be bunched and accelerated. Finally, thephoton. The resulting equation can be written as the system in Eq. (25) has been implemented in a 3D PICsystem code in Sokolov et al., 2009 showing that a laser pulse with intensity 1022 W/cm2 loses about 27% of its energy dxµ 2 IQED eF µν pν  m = pµ + e 2 in the collision with a plasma slab. The same system of   dτ 3 IL m2 equations has been employed to study the penetration µ µ (25) dp dxν p of ultra-intense laser beams into a plasma in the hole- = eF µν    − IQED , dτ dτ m boring regime (Naumova et al., 2009) and to investigatewhere τ is the time in the “momentarily comoving the process of ponderomotive ion acceleration at ultra-Lorentz frame” of the electron where the spatial compo- high laser intensities in overcritical bulk targets (Schlegelnents of pµ instantaneously vanish, IQED is the quantum et al., 2009).radiation intensity (Ritus, 1985) and IL = (2/3)αω02 ξ02 .The expression of IQED in the case of a plane wave isemployed, which is valid only for an ultrarelativistic elec- A. The classical radiation dominated regimetron in the presence of a slowly-varying and undercriticalotherwise arbitrary external field (see Sec. III.B). As it was already observed in Landau and Lifshitz, It has also to be stressed that the original LAD equa- 1975, the fact that the RR force in the LL equation hastion is still the subject of extensive investigation (the to be much smaller than the Lorentz force in the instan-first study of the LAD equation in a plane-wave field was taneous rest-frame of the electron does not exclude thatperformed in Hartemann and Kerman, 1996). In Ferris some components of the two forces can be of the sameand Gratus, 2011, for example, the origin of the Schott order of magnitude in the laboratory system. This occurs 21

if the condition αγ 2 F/Fcr ∼ 1 is fulfilled at any instant, is of the order of the electron’s longitudinal momentumwith γ being the relativistic Lorentz factor of the elec- itself in the laser field. As a result, it is found that intron and F the amplitude of the external electromagnetic the ultrarelativistic case and for a few-cycle pulse, if thefield. For an ultrarelativistic electron this condition can conditionsbe fulfilled also in the realm of classical electrodynam-ics (quantum recoil effects are negligible if γF/Fcr ≪ 1) 4γ02 − ξ02 RC & >0 (28)and it characterizes the so-called Classical Radiation- 2ξ02Dominated Regime (CRDR). The CRDR has been in- are fulfilled, then the electron is reflected in the laservestigated in Shen, 1970 for a background constant and field only if RR is taken into account (see also Harvey anduniform magnetic field. In Koga et al., 2005 an equivalent Marklund, 2012 for a recent investigation of the electron’sdefinition of the CRDR in the presence of a background dynamics in the two complementary regimes 2γ0 ≶ ξ0laser field has been formulated, as the regime where the including RR effects). This can have measurable effectsaverage energy radiated by the electron in one laser pe- if one exploits the high directionality of the radiationriod is comparable with the initial electron energy. By emitted by an ultrarelativistic electron (see Sec. V.A).estimating the radiated power PL from the relativistic The results in Di Piazza et al., 2009a show in fact thatLarmor formula PL = −(2/3)e2 (duµ /ds)(duµ /ds) (Jack- the apex angle of the angular distribution of the emittedson, 1975) with duµ /ds → (e/m)F µν uν , one obtains that radiation, with and without RR effects included, mayPL ∼ αχ0 ξ0 ε0 . Therefore, the conditions of being in the differ by more than 10◦ already at an average opticalCRDR are laser intensity of 5 × 1022 W/cm2 (ξ0 ≈ 150) and at an RC = αχ0 ξ0 ≈ 1, χ0 ≪ 1, (26) initial electron energies of 40 MeV (2γ0 ≈ 156) for which RC ≈ 0.08. Small RR effects on photon spectra emittedwhere, as has been seen in Sec. III.B, the second condi- by initially bound electrons had already been predictedtion ensures in particular that the quantum effects like via numerical integration of the LL equation in Keitelphoton recoil are negligible. The same condition RC ≈ 1 et al., 1998 well below the CRDR.has been obtained in Di Piazza, 2008 by exactly solvingthe LL equation (24) for a general plane-wave backgroundfield. The analytical solution shows in fact that for an B. Quantum radiation reactionultrarelativistic electron the main effect of RR is due tothe last term in Eq. (24). As a consequence, while the The shortcomings of the classical approaches to thequantity u− (φ) is constant if the equation of motion is problem of RR suggest that it can be fully understoodthat due to Lorentz, it decreases here with respect to φ only at the quantum level. In the seminal paper Monizas u− (φ) = u0,− /h(φ), where uµ0 is the four-velocity at and Sharp, 1977 the origin of the classical inconsisten-an initial φ0 and cies, like the existence of runaway solutions of the LAD equation, were clarified in the nonrelativistic case. The 2 2 RC φ authors first show that such inconsistencies are also ab-

dψ(ϕ) Z h(φ) = 1 + dϕ , (27) sent in classical electrodynamics if one considers charge 3 ω 0 φ0 dϕ distributions with a typical radius larger than the classi-where the four-potential of the wave has been assumed cal electron radius r0 = αλC ≈ 2.8 × 10−13 cm. Goingto have the form Aµ (φ) = Aµ0 ψ(φ) (see Sec. III.B, be- to the nonrelativistic quantum theory and by analyzinglow Eq. 10). This effect has been recently suggested in the Heisenberg equations of motion of the electron in anHarvey et al., 2011b as a possible signature to measure external time-dependent field, the authors conclude thatRR (see also Lehmann and Spatschek, 2011). The two the quantum theory of a pointlike particle does not ad-conditions in Eq. (26) are, in principle, compatible for mit any runaway solutions, provided that the externalsufficiently large values of ξ0 . For example, for an opti- field varies slowly along a length of the order of λC (thiscal (ω0 = 1 eV) laser field with an average intensity of is an obvious assumption in the realm of nonrelativis-1024 W/cm2 and for an electron initially counterpropa- tic theory, as time-dependent fields with typical wave-gating with respect to the laser field with an energy of lengths of the order of λC would in principle allow for20 MeV, it is χ0 = 0.16 and RC = 1.3. This example e+ -e− pair production, see Sec. VIII). From this pointshows that, in general, it is not experimentally easy to of view a classical theory of RR has only physical mean-enter the CRDR at least with presently-available laser ing as the classical limit (~ → 0) of the correspondingsystems. In Di Piazza et al., 2009a a different regime has quantum theory and the authors indicate that the result-been investigated, which is parametrically less demand- ing equation of motion is the nonrelativistic LL equationing than the CRDR but in which the effects of RR are with the bare mass m0 (it is shown that the electrostaticstill large. In this regime the change in the longitudi- self-energy of a point charge vanishes in nonrelativisticnal (with respect to the laser field propagation direction) quantum electrodynamics). On the other hand, if onemomentum of the electron due to RR in one laser period considers the quantum equations of motion of a charge 22

distribution and performs the classical limit before the the process of multiphoton Compton scattering and then,pointlike limit, then the classical equations of motion of classically, to the Lorentz dynamics. Whereas, all high-the charge distribution are, of course, recovered and, once order terms give rise to radiative corrections and to high-the point-like limit is then performed, runaway solutions order coherent and incoherent (cascade) processes, andappear again. The nonrelativistic form of the LL equa- determine what we call “quantum RR”. Here, by high-tion has been also recovered from quantum mechanics order coherent processes is meant those involving morein Krivitskii and Tsytovich, 1991 by including radiative than one basic QED process (photon emission by ancorrections to the time-dependent electron momentum electron/positron or e+ -e− photoproduction) but all oc-operator in the Heisenberg representation, and by calcu- curring in the same formation region. Analogously, inlating the time-derivative of the average momentum in a higher-order incoherent or cascade processes each basicsemiclassical state. QED process occurs in a different formation region. Now, The situation in the relativistic theory is less straight- in the case of a background plane wave at ξ0 ≫ 1 andforward because relativistic quantum electrodynamics, χ0 . 1, the quantum effects are certainly important buti.e., QED, is a field theory fundamentally different from the radiative corrections and higher-order coherent pro-classical electrodynamics. The first theory of relativis- cesses scale with α and can be neglected (Ritus, 1972).tic quantum RR goes back to W. Heitler and his group Also, if χ0 does not exceed unity then the photons emit-(Heitler, 1984; Jauch and Rohrlich, 1976). However, the ted by the electron are mainly unable, by interactingevaluations of the QED amplitudes in Heitler’s theory in- again with the laser field, to create e+ -e− pairs, as thevolve the solution of complicated integral equations and pair production probability is exponentially suppressedit has given a practical result only in the calculation of (see also Secs. VIII and IX). Therefore, it can be con-the total energy emitted by a nonrelativistic quantum cluded that at ξ0 ≫ 1 and χ0 . 1, RR in QED corre-oscillator, with and without RR. sponds to the overall photon recoil experienced by the At first sight one would say that RR effects are auto- electron when it emits many photons consecutively andmatically taken into account in QED, because the elec- incoherently (Di Piazza et al., 2010a).tromagnetic field is treated as a collection of photons A qualitative understanding of the above conclusionthat take away energy and momentum, when they are can be attained by assuming that χ0 ≪ 1 and by esti-emitted by charged particles. However, photon recoil is mating the average number Nγ of photons emitted byalways proportional to ~, making it a purely quantum an electron in one laser period at ξ0 ≫ 1. Since thequantity with no classical counterpart. Moreover, if one probability of emitting one photon in a formation lengthcalculates the spectrum of multiphoton Compton scat- is of the order of α and since one laser period containstering in an external plane-wave field, for example, and about ξ0 formation lengths (Ritus, 1985) then Nγ ∼ αξ0 .then performs the classical limit χ0 → 0, one obtains Also, the typical energy ω ′ of a photon emitted by anthe corresponding multiphoton Thomson spectrum cal- electron is of the order ω ′ ∼ χ0 ε0 , then the average en-culated via the Lorentz equation and not via the LAD ergy E emitted by the electron is E ∼ αξ0 χ0 ε0 = RC ε0 .or the LL equation (see also Sec. V.A). Finally, it has This estimate is in agreement with the classical result ob-also been seen that in classical electrodynamics the RR tained from the LL equation. In other words, the classicaleffects may not be a small perturbation on the Lorentz limit of RR in this regime corresponds to the emission ofdynamics and they cannot be obtained as the result of a a higher and higher number of photons all with an en-single limiting procedure. Otherwise they would always ergy much smaller than the electron energy, in such aappear as a small correction. way that even though the recoil at each emission is al- In order to understand what RR is in QED, it is more most negligible, the cumulative effect of all photon emis-convenient to go back to Eq. (21) and to notice that sions may have a finite nonnegligible effect. Note that ω ′the LAD, namely the LL, equation is equivalent to the and Nγ are both pure quantum quantities and only theircoupled system of Maxwell and Lorentz equations. If one product E has a classical analogue in the limit χ0 → 0.determines the trajectory of the electron via the LL equa- These considerations have allowed for the introductiontion and then calculates the total electromagnetic field in Di Piazza et al., 2010a of the Quantum Radiation-FTµν (x) via the Liénard-Wiechert four-potential (Landau Dominated Regime (QRDR), which is characterized byand Lifshitz, 1975), one has solved completely the classi- multiple emission of photons already in one laser period.cal problem of the radiating electron in the given electro- This regime is then characterized by the conditionsmagnetic field. As has been discussed in Sec. III.B, the RQ = αξ0 ≈ 1, χ0 & 1. (29)solution of the analogous problem in strong-field QEDwould correspond to completely determine the S-matrix Quantum photon spectra have been calculated numeri-in Eq. (9), as well as the asymptotic state |t → +∞i cally in Di Piazza et al., 2010a without RR, i.e., by in-for the given initial state |t → −∞i = |e− i, which rep- cluding only the emission of one photon (four-momentumresents a single electron. The first-order term in the k ′µ ), and with RR, i.e., by including multiple-photonperturbative expansion of the S-matrix corresponds to emissions (and by integrating with respect to all the four- 1

tromagnetic field in the vacuum itself. The possibility of

photon-photon interaction in vacuum, within the frame- work of QED, can be understood qualitatively by observ- ing that a photon may locally “materialize” into an e+ -e− pair which, in turn, interacts with other photons. For the same reason a background electromagnetic field can in- fluence photon propagation (see Fig. 12 and Berestetskii et al., 1982). In the latter case the extension lf of the 23

ground electromagnetic ﬁeld. The thick electron lines indicate

electron propagators calculated in the Furry picture, account- ing exactly for the presence of the background ﬁeld.

region where this transformation occurs, i.e., its forma-

tion length, depends, in principle, on the structure ofmomenta of the emitted photons except one indicated as the background field (Baier and Katkov, 2005). How-k ′µ ). The results show that in the QRDR the effects of ever, in some cases it can be estimated qualitatively viaRR are essentially three (see Fig. 11): 1) increase of the the Heisenberg uncertainty principle from the typical mo-photon yield at low photon energies; 2) decrease of the mentum p flowing in the e+ -e− loop in Fig. 12. We con-photon yield at high photon energies; 3) shift of the max- sider, for example, a constant background electromag-imum of the photon spectrum towards low photon ener- netic field (or a slowly-varying one, at leading order ingies. Figure 11 also shows that the classical treatment of the space-time derivatives of the field itself). In this case,RR (via the LL equation) artificially overestimates the if the energy ω of the incoming photon (see Fig. 12) is atabove effects, the reason being that quantum corrections most of the order of m, then the momentum p flowing indecrease the average energy emitted by the electron with the e+ -e− loop is of the order of m and lf ∼ 1/p ∼ λC .respect to the classical value (Ritus, 1985). However, at If ω ≫ m the analysis is more complicated and the for-χ0 ≪ 1, i.e., when the recoil of each emitted photon is mation length strongly depends on the structure of themuch smaller than the electron energy, then the quantum background field.spectra converge into the corresponding classical ones. From the theoretical point of view it is convenient toAs mentioned in Sec. VI, a semiclassical phenomenolog- distinguish between low-energy vacuum-polarization ef-ical approach to RR in the quantum regime has been fects if ω ≪ m and high-energy ones if ω & m.proposed in Sokolov et al., 2009, 2010b. Finally, the quantum modifications induced by theelectron’s self-field onto the Volkov states (see Eq. (6)) A. Low-energy vacuum-polarization effectshave been recently investigated in Meuren and Di Pi-azza, 2011. It is found that the classical expression of The scattering in vacuum of a real photon by anotherthe electron quasimomentum q0µ in a linearly polarized real photon is possibly the most fundamental vacuum-plane wave (see Eq. (8)) admits a correction depend- polarization process (Berestetskii et al., 1982) and iting on the quantum parameter χ0 and also that self-field has not yet been observed experimentally. The totaleffects induce a peculiar dynamics of the electron spin. cross section of the process depends only on the Lorentz- invariant parameter η = (k1 k2 )/m2 , with k1µ and k2µ be- ing the four-momenta of the colliding photons or, equiv-VII. VACUUM-POLARIZATION EFFECTS alently, on the energy ω ∗ of the two colliding photons in their center-of-momentum system (η = 2ω ∗ 2 /m2 ). This QED predicts that photons interact with each other process has been investigated in Euler, 1936 in the low-also in vacuum (Berestetskii et al., 1982). Effects aris- energy limit η ≪ 1 and then in Akhiezer, 1937 in theing from this purely quantum interaction are referred to high-energy limit η ≫ 1. The complete expression of theas vacuum polarization effects. This is in contrast to cross section σγγ→γγ was calculated in Karplus and Neu-classical electrodynamics where the linearity of Maxwell’s man, 1950 and can also be found in Berestetskii et al.,equations in vacuum forbids self-interaction of the elec- 1982 (see also Fig. 13). In the low-energy limit η ≪ 1 24

1 108 13 2 148 4 σγγ→γγ = + π − 8π 2 ζ(3) + π 1. Experimental suggestions for direct detection of π 5 2 225 photon-photon scattering η≫1 4 2 1 i − 24ζ(5) α λC , η The most direct way to search for photon-photon scat- (31) tering events in vacuum by means of laser fields is to let two laser beams collide and to look for scattered photons.where ζ(x) is the Riemann zeta function (Olver et al., However, by employing a third “assisting” laser beam, if2010). In terms of the center-of-momentum energy one of the final photons is kinematically allowed to beω ∗ , the cross section becomes σγγ→γγ [cm2 ] = 7.4 × emitted along this beam with the same frequency and po-10−66 (ω ∗ [eV])6 at ω ∗ ≪ m and σγγ→γγ [cm2 ] = 5.4 × larization, then the number of photon-photon scattering10−36 /(ω ∗ [GeV])2 at ω ∗ ≫ m. The steep dependence events can be coherently enhanced (Varfolomeev, 1966).of σγγ→γγ on η for η ≪ 1 is the main reason why real In this laser-assisted setup the “signal” of photon-photonphoton-photon scattering has, so-far, eluded experimen- scattering is, of course, the remaining outgoing photon.tal observation (see the reviews Marklund and Shukla, In Lundin et al., 2007 and Lundström et al., 2006 an2006 and Salamin et al., 2006 for experiments and ex- experiment has been suggested to observe laser-assistedperimental proposals until 2005 aiming to observe real photon-photon scattering with the Astra-Gemini laserphoton-photon scattering in vacuum). system (see Sec. II.A). The authors found a particular However, various proposals have been put forward re- “three-dimensional” setup, which turns out to be espe-cently in order to observe this process by colliding strong cially favorable for the observation of the process (seelaser beams which contain a large number of photons. Fig. 14). The number Nγ of photons scattered in oneA common theoretical starting point of all these pro- shot for an optimal choice of the geometrical factors andposals is the “effective-Lagrangian” approach (Dittrich of the polarization angles between the incoming and theand Gies, 2000; Dittrich and Reuter, 1985). In this ap- assisting beams is found to beproach the interaction among photons in vacuum is de-scribed via an effective Lagrangian density of the elec- P1 [PW]P2 [PW]P3 [PW]tromagnetic field. By starting from the total Lagrangian Nγ ≈ 0.25 . (33) (λ4 [µm])3density of the classical electromagnetic field and of thequantum e+ -e− Dirac field, one integrates out the de- Here, P1 and P2 are the powers of the incoming beams,grees of freedom of the latter field and is left with a P3 is the power of the assisting beam and λ4 is the wave-Lagrangian density depending only on the electromag- length of the scattered wave to be measured. By plug-netic field. As has been seen above, the formation re- ging in the feasible values for Astra-Gemini P1 = P2 =gion of photon-photon interaction at low energies is of 0.1 PW and P3 = 0.5 PW, one obtains Nγ ≈ 0.07, i.e., 25

FIG. 15 (Color) Intensity Id of the vacuum-scattered wave

for a probe beam propagating along the positive y direction and colliding with two strong beams aligned along the x axis.FIG. 14 (Color) Schematic “three-dimensional” setup for The crosses correspond to coordinates xn according to thelaser-assisted photon-photon scattering involving two incom- classical prediction xn = (n + 1/2)λp d/D, where n in an inte-ing beams (in blue), an assisting one (in red) and a scattered ger number, λp is the wavelength of the probe ﬁeld, d is theone (in violet). From Lundström et al., 2006. distance between the interaction region and the observation screen and D is the distance between the centers of the two strong beams. The numerical values of the parameters can beroughly one photon scattered every 15 shots (for the pa- found in King et al., 2010a. Adapted from King et al., 2010a.rameters of Astra Gemini the wavelength of both incom-ing beams is chosen as 0.4 µm, that of the assisting beam gating beams. For optimal lasers parameters and at aas 0.8 µm, so that the wavelength λ4 = 0.276 µm of thescattered photon is different from those of the incoming strong-laser power of 100 PW the diffracted vacuum sig-and assisting beams). nal is predicted to be measurable in a single shot. The effects on photon-photon scattering of the temporal pro- The quantum interaction among photons in vacuum file of the laser pulses have been recently investigatedhas been exploited in King et al., 2010a to propose, for in King and Keitel, 2012, showing a suppression of thethe first time, a double-slit setup comprised only of light number of vacuum scattered photons with respect to the(see also Marklund, 2010). In this setup two strong par- infinite-pulse (monochromatic) case.allel beams collide head-on with a counterpropagating The concept of Bragg scattering has been exploited inprobe pulse. The photons of the probe have the choice Kryuchkyan and Hatsagortsyan, 2011 to observe the scat-to interact either with one or with the other strong beam, tering of photons by a modulated electromagnetic-fieldand, when scattered, they are predicted to build an in- structure in vacuum. If a probe wave passes throughterference pattern with alternating minima and maxima a series of parallel strong laser pulses and if the Braggtypical of double-slit experiments (see Fig. 15). Also, condition on the impinging angle is fulfilled, the num-if one of the slits is closed, i.e., if the probe collides ber of diffracted photons can be strongly enhanced. Atwith only one strong beam, the interference fringes dis- a fixed intensity for each strong beam the enhancementappear. The key idea behind this setup is that the factor with respect to laser-assisted photon-photon scat-vacuum-scattered beam (intensity Id ), although propa- tering is equal to the number of beams in the periodicgating along the same direction as the probe, has a much structure. However, in experiments usually the total en-wider angular distribution than the latter, offering the ergy of the laser beams is fixed and an enhancement bypossibility of detecting vacuum-scattered photons out- a factor of two is predicted. By considering NG equalside the focus of the undiffracted probe beam. For a Gaussian pulses propagating along the x direction andstrong-field intensity of I0 ≈ 5 × 1024 W/cm2 that may their centers separated by a distance D > 2wz from eachbe in the near future be available at ELI or at HiPER other, the resulting photon-photon scattering probabil-and for a probe beam with wavelength λp = 0.527 µm ity will be proportional to the phase-matching factor P,and intensity Ip = 4 × 1016 W/cm2 , it is predicted that withabout four photons per shot would contribute to build upthe interference pattern in the observable region (it is the sin2 (δkz NG D/2)region outside the circle in Fig. 15, where Id > 100 Ip ). P= , (34) sin2 (δkz D/2)The diffraction of a probe beam in vacuum by a single fo-cused strong laser pulse is also investigated in Tommasini where the vector δk = k2 − k1 is the difference betweenand Michinel, 2010 in the case of almost counterpropa- the wave vectors of the reflected and incident waves. The 26

ing an integer. By employing ten optical laser beams The birefringence of the polarized vacuum is exploitedwith a wavelength of 1 µm and each with an intensity in Heinzl et al., 2006 to show that if a linearly-polarizedof 2.3 × 1023 W/cm2 , about five vacuum-scattered pho- probe x-ray beam (wavelength λp ) propagates along atons are predicted per shot. Finally, an enhancement strong optical standing wave, then it emerges from theof vacuum-polarization effects in laser-laser collision has interaction elliptically polarized with ellipticity ǫ givenbeen predicted in Monden and Kodama, 2011 by em- byploying strong laser beams with large angular aperture. 2α l0,R I0For example, it is predicted that the number of vacuum- ǫ= κ , (39)radiated photons will be enhanced by two orders of mag- 15 λp Icrnitude, if the angular aperture of the colliding beams is where κ ∼ 1 is a geometrical factor and l0,R is theincreased from 53◦ to 103◦ . Rayleigh length of the intense laser beam. If this beam Other experimental suggestions to measure photon- is generated by a laser like ELI (I0 ∼ 1025 W/cm2 ), val-photon scattering in vacuum can be found in Eriksson ues of the ellipticities of the order of 10−7 are predictedet al., 2004 and Tommasini et al., 2008. at λp = 0.1 nm. Recent advances on x-ray polarime- try allow for measurement of ellipticities of the order of 10−9 at a wavelength of 0.2 nm (Marx et al., 2011). In2. Polarimetry-based experimental suggestions Ferrando et al., 2007 a phase-shift has been found the- oretically to be induced by vacuum-polarization effects The expression of the Euler-Heiseberg Lagrangian den- when two laser beams cross in the vacuum, which is pre-sity in Eq. (32), suggests to interpret a region where only dicted to be measurable at laser intensities available atan electromagnetic field is present as a material medium ELI or at HiPER.characterized by a polarization PEH = ∂LEH /∂E − When an electromagnetic wave with wavelength λ im-E/4π and a magnetization MEH = ∂LEH /∂B + B/4π pinges upon a material body, the features of the scattered(Jackson, 1975) given by radiation depend on the so-called diffraction parameter α D = l⊥ 2 /λd (Jackson, 1975). Here, l⊥ is the spatial di- 2(E 2 − B 2 )E + 7(E · B)B , (35)

MEH = 2 2 180π Fcr the screen, where the radiation is detected, from the in- teraction region. The near region, D ≫ 1, is known asNote that an arbitrary single plane wave cannot “polar- the “refractive-index limit”, because the effects of theize” the vacuum, as in this case PEH and MEH identi- presence of the body can be described as if the wavecally vanish. Equations (35) and (36) indicate that the propagates through a medium with a given refractivepresence of an electromagnetic field in the vacuum alters index. However, if D . 1 then the diffraction effectsthe vacuum’s refractive index. The situation is even more become important and description of the wave-body in-complicated because of the vectorial nature of the back- teraction only in terms of a refractive index is in gen-ground electromagnetic field which polarizes the vacuum eral not possible. This aspect has been pointed out inand introduces a privileged direction in it. As a result, Di Piazza et al., 2006 within the context of light-lightthe vacuum’s refractive index is altered in a way that interaction in vacuum (see also Di Piazza et al., 2007a).depends, in general, on the mutual polarizations of the Tight focusing required to reach high intensities usuallyprobe electromagnetic field and of the background field: renders the interaction region so small that diffractionthe polarized vacuum behaves as a birefringent medium. effects may become substantial at typical experimentalFor example, in the case of an arbitrary constant electro- conditions. In some cases diffractive effects reduce by anmagnetic field (E, B), the refractive indices nEH,1/2 of order of magnitude the values of the ellipticity calculateda wave propagating along the direction n and polarized via the refractive-index approach and also induce a ro-along one of the two independent directions u1 = E/|E| tation of the main axis of the polarization ellipse withand u2 = B/|B|, with E = E − (n · E)E + n × B and respect to the initial polarization direction of the probeB = B − (n · B)B − n × E are given by (Dittrich and field (Di Piazza et al., 2006). Tight focusing of the strongGies, 2000) polarizing beam requires quite a detailed mathematical 4α (n × E)2 + (n × B)2 − 2n · (E × B) description employing a realistic focused Gaussian beam,nEH,1 = 1 + 2 , while a simpler description was employed for the usually 90π Fcr (37) weakly-focused probe beam. This prevented the appli- cability of the results in the so-called far region where 7α (n × E)2 + (n × B)2 − 2n · (E × B)nEH,2 = 1 + , D ≪ 1 and where the spatial spreading of the probe field 90π 2 Fcr is also important. This assumption has been recently re- (38) moved in King et al., 2010b, where it was also pointed 27

out that by considering the diffraction of a probe beam where

by two separated beams instead of that by a single stand- v u !ing wave, an increase in the ellipticity and in the rotation u 4πe2 ni 1 Z n0 = t 1 − +p ,of polarization angle by a factor 1.5 is expected. mω02 p 1 + ξ02 (mi /m)2 + Z 2 ξ02 A different method based on the phase-contrast Fourier (41)imaging technique has been suggested in Homma et al., is the refractive index of the plasma without vacuum-2011 to detect vacuum birefringence. This technique pro- polarization effects (Mulser and Bauer, 2010). Equationvides a very sensitive tool to measure the absolute phase (40) already indicates the possibility of enhancing the ef-shift of a probe beam when it crosses an intense laser fects of vacuum polarization by working at laser frequen-field. Numerical simulations demonstrate the feasibility cies ω0 such that n0 ≪ 1, i.e., close to the effective plasmaof measuring vacuum birefringence also by employing an critical frequency. This region of parameters is in generaloptical probe field and a 100-PW strong laser beam. complex to investigate, due to the arising of different in- Photon “acceleration” in vacuum due to vacuum polar- stabilities. However, the idealized situation investigatedization has been studied in Mendonça et al., 2006. This in Di Piazza et al., 2007b shows, at least in principle, theeffect corresponds to a shift of the photon frequency when possibility of enhancing the vacuum-polarization effectsit passes through a strong electromagnetic wave. If kpµ by an order of magnitude at a given intensity I0 withis the four-momentum of a probe photon with energy respect, for example, to the results in Di Piazza et al.,ωp when it enters a region where a strong laser beam 2006. The effects of the presence of an additional strongis present, then, due to vacuum-polarization effects, ωp constant magnetic field have been analyzed in Lundinbecomes sensitive to the gradient of the intensity of the et al., 2007. The general theory presented in this pa-strong beam. As a result, a frequency up-shift (down- per covers different waves propagating in a plasma asshift) is predicted at the rear (front) of the strong beam. Alfvén modes, whistler modes, and large-amplitude laser According to Eqs. (37) and (38) the phase velocity modes. We also mention the recent paper Bu and Ji,of light in vacuum is smaller than unity. This circum- 2010, in which the photon acceleration process has beenstance has been exploited in Marklund et al., 2005, where investigated in a cold plasma and the reference BrodinCherenkov radiation by ultrarelativistic particles mov- et al., 2007, where vacuum-induced photon splitting ining with constant velocity in a photon gas has been pre- a plasma is studied. Finally, we mention the possibilitydicted, if the speed of the particle exceeds the phase ve- of testing nonlinear vacuum QED effects in waveguides.locity of light. Finally, in Zimmer et al., 2012 an induced In Brodin et al., 2001 the generation of new modes inelectric dipole moment of the neutron has been proposed waveguides due to vacuum-polarization effects has beenas a signature of the polarization of the QED vacuum. predicted. More recently signatures of nonlinear QED effects in the transmitted power along a waveguide have3. Low-energy vacuum-polarization effects in a plasma been analyzed in Ferraro, 2010.

Equations (37) and (38) indicate that vacuum-

B. High-energy vacuum-polarization effectspolarization effects elicited by a plane wave with intensityI0 would alter the vacuum refractive index by an amount Generally speaking the treatment of vacuum-of the order of (α/45π)(I0 /Icr ). It was first realized in polarization effects for an incoming photon with energyDi Piazza et al., 2007b that this aspect can be in prin- ω in the presence of a background electromagnetic fieldciple significantly improved in a plasma. For the sake with a typical angular frequency ωb cannot be performedof simplicity the case of a cold plasma was considered in an effective Lagrangian approach if ωωb /m2 & 1: theand the vacuum-polarization effects were implemented in incoming photon “sees” the nonlocality of its interactionthe inhomogeneous Maxwell’s equations as an additional with the background electromagnetic field through its“vacuum four-current” (Di Piazza et al., 2007b). Now, local “transformation” into an e+ -e− pair (see Fig. 12).unlike in the vacuum, the field invariant F (x) for a single The technical difficulty in treating vacuum-polarizationmonochromatic circularly polarized plane wave does not effects at high energies arises from the fact that thevanish in a plasma. Thus, vacuum-polarization effects in interaction between the virtual e+ -e− pair and thea plasma already arise in the presence of a single travel- background field has to be accounted for exactly. Thising plane wave. In Di Piazza et al., 2007b the vacuum- has been accomplished for the background field of acorrected refractive index n of a two-fluid electron-ion nucleus with charge number Z such that Zα ∼ 1 andplasma (ion mass, density and charge number given by Delbrück scattering and photon splitting in such a fieldmi , ni and Z, respectively) in the presence of a circularly have also been observed experimentally (see the reviewspolarized plane wave has been found as r Lee et al., 2003 and Milstein and Schumacher, 1994). 2α I0 As we have recalled in Sec. III.B, the Dirac equa- n = n20 + (1 − n20 )2 , (40) tion in a background plane wave described by the four- 45π Icr 28

vector potential Aµ (φ) can be solved exactly and ana- proton collides head-on with a strong laser field. Thelytically. Accordingly, the exact electron (Volkov) prop- quantum interaction of the Coulomb field of the protonagator G(x, y|A) in the same background field has also with the laser field allows for a merging of laser photonsbeen determined (see, e.g., Ritus, 1985). The so-called into a single high-energy photon. The use of a proton,“operator technique”, developed in Baier et al., 1976a,b, instead of an electron, for example, is required in order toturns out to be very convenient for investigating vacuum- suppress the background process of multiphoton Thom-polarization effects at high energies (the operator tech- son/Compton scattering, where again many photons ofnique for a constant background field was developed the laser can be directly absorbed by the proton and con-in Baier et al., 1975a,b; and Schwinger, 1951). In verted into a single high-energy photon (see Sec. V.A).this technique a generic electron state Ψ(x) in a plane- In fact, the kinematics of the two processes, vacuum-wave field and the propagator G(x, y|A) are intended mediated laser photon merging and multiphoton Thom-as the configuration representation of an abstract state son/Compton scattering is the same, except that in the|Ψi and of an operator G(A) such that Ψ(x) = hx|Ψi treatment in Di Piazza et al., 2008a only an even num-and G(x, y|A) = hx|G(A)|yi. In particular, since the ber of laser photons can merge in the vacuum-mediatedpropagator G(x, y|A) is the solution of the equation process. The probability of ℓ-photon Thomson/Compton{γ µ [i∂µ − eAµ (φ)] − m}G(x, y|A) = δ(x − y), then the scattering of a particle with charge Q and mass M de-abstract operator G(A) is simply pends on the parameter ξ0,c = |Q|E0 /M ω0 and scales 2ℓ as ξ0,c at ξ0,c ≪ 1. Thus, the use of a “heavy” particle 1 G(A) = , (42) like a proton (mass mp = 1.8 × 103 m = 938 MeV) is γ µ [Pµ − eAµ (φ)] − m essential to suppress this background process. Note thatwith P µ being the four-momentum operator. Evaluation in order to have ξ0,p = |e|E0 /mp ω0 ≈ 1 for a proton,via the operator technique of the matrix element corre- a laser field with intensity of the order of 1024 W/cm2sponding to a generic vacuum-polarization process is then is required. It is found that for an optical backgroundcarried out by manipulating abstract operators, which is field such that ξ0 ≫ 1, the amplitude of the (2ℓ)-photoneasier than by working with the corresponding quantities merging process depends only on the nonlinear quantumin configuration space. parameter In Di Piazza et al., 2007c the operator technique has E0 2ℓ(1 + vp )ω0 1 − cos ϑ (2ℓ)been employed to calculate the rate of photon splitting χ0,p = , (43) Fcr m 1 + vp cos ϑin a strong laser field for an incoming photon with four-momentum k µ . This was the first investigation of a QED where vp is the proton velocity and ϑ is the angle betweenprocess involving three Volkov propagators. The calcu- the direction of the emitted photon and the propagationlated rate is valid for an arbitrary plane-wave field, pro- direction of the plane wave. By colliding a proton beam,vided that radiative corrections can be neglected, i.e., at of energy available at the Large Hadron Collider (LHC) 2/3 of the order of 7 TeV (LHC, 2011) with an optical laserακ0 ≪ 1, with κ0 = (k− /m)(E0 /Fcr ), in the most un-favorable regime ξ0 , κ0 ≫ 1 (Ritus, 1972). In fact, as we beam of intensity of 3 × 1022 W/cm2 , it is demonstratedhave mentioned in Sec. III.B, QED processes involving that the vacuum-mediated merging of two laser photonsthe collision of a photon and an intense plane wave are and the analogous two-photon Thomson/Compton scat-controlled by the two Lorentz- and gauge-invariant pa- tering have comparable rates, implying that the inclu-rameters ξ0 and κ0 . In Di Piazza et al., 2007c it turned sive signal should be twice the one expected withoutout to be more convenient to perform a parametric study vacuum-polarization effects. By accounting for the de-of the photon-splitting rate by varying the two parame- tails of the proton beams available at LHC and of theters ξ0 and η0 = ω0 k− /m2 (note that κ0 = η0 ξ0 ). By laser system PFS (see Sec. II.A), about 670 two-photonemploying the Furry theorem (Berestetskii et al., 1982), merging events and about 5 four-photon merging eventsit is shown that photon splitting in a laser field only oc- are expected per hour. In the discussed setup most of thecurs with absorption of an odd number of laser photons. photons are emitted almost in the direction of the proton (2)In particular, if the strong field is circularly polarized and velocity (ϑ ≈ π) in which χ0,p ∼ 1. It is also indicated inif it counterpropagates with respect to the incoming pho- Di Piazza et al., 2008a that the use of the perturbativeton, conservation of the total angular momentum along expression of the laser-photon merging rate at leading (2)the propagation direction of the beams implies that for order in χ0,p ≪ 1 would lead to an error of about 30%.η0 ≪ 1 photon splitting can occur only via absorption of Other setups for observing vacuum-polarization effects inone or of three laser photons. laser-proton collisions have been discussed in Di Piazza In Di Piazza et al., 2008a a physical scenario has been et al., 2008b, involving, for example, XFEL or single in-advanced in which nonperturbative vacuum-polarization tense XUV pulses. Finally, the process of Delbrück scat-effects can be in principle observed (here “nonperturba- tering in a combined Coulomb and laser field has beentive” means “high-order” in the quantum nonlinearity studied in Di Piazza and Milstein, 2008. Here an incom-parameter, see below). In this scenario a high-energy ing photon is scattered by the Coulomb field of a nucleus

29

a) b) c) These processes share the common feature to possess

x different interaction regimes which are mainly character- ized by the value of the parameter ξ0 . When ξ0 ≪ 1 the presence of the laser field can be taken into accountFIG. 16 Feynman diagrams corresponding to processes (i) perturbatively and this yields a pair production rate R(part a)), (ii) (part b)) and (iii) (part c)), respectively. The of the form Re+ -e− ∼ mξ02ℓm , where ℓm is the minimumthick continuous lines in parts a) and b) indicate Volkovpositive- and negative-energy states. The crossed vertex in integer number which kinematically allows the process.part b) stands for the Coulomb electromagnetic ﬁeld. The For the process (i) it is ℓm (k0 k) > 2m2 , with k0µ and k µdiagram in part c) is related to the vacuum current jvac µ (x), being the four-momentum of the laser photon and the + −that one has to determine in order to calculate the e -e pair incoming photon, respectively. For the process (ii) it isyield (Dittrich and Gies, 2000). The double line indicates the ℓm ω0⋆ > 2m, where ω0⋆ = ω0 uc,− is the laser angular fre-electron propagator calculated in the Furry picture including quency in the rest-frame of the charge which produces theexactly the background standing wave. Coulomb field and which has four-velocity uµc . Finally, for the process (iii) it is ℓm ω0 > 2m. Due to the specific dependence of the pair production rate on ξ0 , this regimeand by a strong laser field. While the presence of the of pair production is called multiphoton regime. In con-laser field is taken into account exactly in the calcula- trast, when ξ0 ≫ 1 the presence of the laser field has totions, only leading-order effects in the nuclear parameter be taken into account exactly by performing the calcu-Zα are accounted for. Analogously to Di Piazza et al., lations in the Furry picture. As the condition ξ0 ≫ 12008a, it is found that high-order nonlinear corrections is realized for vanishing laser frequencies at a fixed laserin the parameter κ0 to the cross section of the process amplitude, this regime is called quasistatic regime andalready become important at κ0 ≈ 0.2. For example, the pair-production rate here is governed by a differentthese corrections amount to about 50% at κ0 = 0.35. parameter which depends on the process at hand. For process (i), for example, the form of the rate depends on the physical parameter κ0 introduced in Sec. III.BVIII. ELECTRON-POSITRON PAIR PRODUCTION 3/2 and it scales as ∼ mκ0 exp(−8/3κ0 ) if κ0 ≪ 1 and as 2/3 One of the most important predictions of QED has ∼ mκ0 if κ0 ≫ 1 (Nikishov and Ritus, 1964a; Reiss,been the possibility of transforming light into matter 1962). For process (ii), we distinguish the case in which(Dirac, 1928). If two photons with four-momenta k1µ the incoming particle is an electron, from that in whichand k2µ collide at an angle such that the parameter η = it is a heavier particle like a nucleus with charge num-(k1 k2 )/m2 exceeds two, the creation of an e+ -e− pair be- ber Z. In the first case the pair production rate dependscomes kinematically allowed (Breit-Wheeler e+ -e− pair on the parameter χ0 , already introduced in Sec. III.B,production (Breit and Wheeler, 1934)). Shortly after the and the recoil due to the pair creation on the electronrealization of the laser in 1960, theoreticians started to has to be taken into account (see also Sec. VIII.A). Instudy possibilities for the creation of e+ -e− pairs from the second case the motion of the nucleus is usually as-vacuum by very strong laser fields (Nikishov and Ritus, sumed not to be altered by the pair creation process and1964a; Reiss, 1962; Yakovlev, 1965). Because of con- the nucleus itself is described as a background Coulombstraints from energy-momentum conservation, a single field (see also Sec. VIII.B). The pair-production rateplane-wave laser field cannot create pairs from vacuum, depends on the parameter χ0,n = un,− (E0 /Fcr ), withno matter how intense it is. For a single plane wave, in uµn being the four-velocity of the nucleus, and on thefact, all the photons propagate along the same direction nuclear parameter Zα. Specifically, √ the pair-productionand the parameter η vanishes identically for any pair of rate scales as m(Zα)2 exp(−2 3/χ0,n ) if χ0,n ≪ 1 andphotons in the plane wave. Thus, an additional source of as m(Zα)2 χ0,n ln χ0,n if χ0,n ≫ 1 (Milstein et al., 2006;energy is therefore required to trigger the process of pair Yakovlev, 1965). Finally, for process (iii), the rate Re+ -e−production in a plane wave. There are essentially three has been derived mainly by approximating the stand-different possibilities (see also Fig. 16): ing wave as an oscillating electric field (see also Sec. VIII.C). It is found that Re+ -e− depends on the ratio (i) pair production by a high-energy photon propa- Υ0 = E0 /Fcr , with E0 being the amplitude of the stand- gating in a strong laser field (multiphoton Breit- ing wave in the (fixed) laboratory frame, and that it Wheeler pair production); scales as mΥ20 exp(−π/Υ0 ) if Υ0 ≪ 1 and as mΥ20 if Υ0 ≫ 1 (Brezin and Itzykson, 1970; Popov, 1971, 1972). (ii) pair production by a Coulomb field in the presence As expected, these scalings coincide with the correspond- of a strong laser field; ing ones in a constant electric field E0 (Schwinger, 1951).

(iii) pair production by two counterpropagating strong The physical meaning of the three parameters κ0 , χ0,n laser beams forming a standing light wave. and Υ0 can be qualitatively understood in the follow- 30

ing way. For process (i) the dressing of the electron A. Pair production in photon-laser and electron-laserand positron mass (see Sec. III.B) modifies the thresh- collisionsold of e+ -e− pair production at ξ0 ≫ 1 according toℓm (k0 k) & 2m2 ξ02 . Now, analogously to multiphoton Among the pair-production processes mentionedThomson and Compton scattering (see Sec. V.A), the above, only laser-induced pair production for ξ0 < 1 hastypical number of laser photons absorbed in pair pro- been observed experimentally. Its feasibility has beenduction via photon-laser collision is of the order of ξ03 shown, in fact, in the pioneering E-144 experiment at(Nikishov and Ritus, 1964a) and the threshold condition SLAC (Bamber et al., 1999; Burke et al., 1997) (seebecomes κ0 & 1. Concerning the process (ii), the ap- Reiss, 1971 for a corresponding theoretical proposal).pearance of the parameter χ0,n in the quasistatic limit The experiment relied on collisions of the 46.6 GeV elec-ξ0 ≫ 1 can be understood by noting that the quantity tron beam from SLAC’s linear accelerator with a coun-un,− E0 is the amplitude of the laser field in the rest-frame terpropagating intense laser pulse of photon energy ofof the nucleus and that a constant and uniform electric ω0 = 2.4 eV and intensity 1.3 × 1018 W/cm2 (ξ0 ≈ 0.4).field with strength of the order of Fcr = m2 /|e| supplies In the rest-frame of the electrons, the laser intensity anda e+ -e− pair with its rest energy 2m along the typical frequency are largely Doppler up-shifted to the requiredlength scale of QED λC = 1/m (see also Sec. III.B). level and the pair generation probability is effectively en-This last observation also demonstrates the presence of hanced. In principle both reactions (i) and (ii) contributethe parameter Υ0 for process (iii). The typical exponen- to pair production in this kind of collisions. Based ontial scaling of the pair production rate for ξ0 ≫ 1, and separate simulations of both production channels, reac-at κ0 ≪ 1, χ0,n ≪ 1 and Υ0 ≪ 1 for processes (i), (ii) tion (i) was found to dominate. The high-energy photonand (iii), respectively, is reminiscent of a quantum tun- originates from multiphoton Compton backscattering ofneling process. Thus, one refers to this regime also as a laser photon off the electron beam.tunneling pair production (note, however, that the no- Despite the significance of the SLAC experiment, a uni-tion of “tunneling” in laser-induced processes should be fied description of pair creation in electron-laser collisionsregarded with special care beyond the dipole approxima- has been presented only recently, treating the competingtion (Klaiber et al., 2012; Reiss, 2008)). mechanisms (i) and (ii) within the same formalism (Hu In all processes discussed above, the laser field is al- et al., 2010). Good agreement with the experimental re-ways participating directly in the pair creation step and sults has been obtained. Moreover, it was shown that thefundamental properties of the quantum vacuum under SLAC study observed the onset of nonperturbative pairextreme high-field conditions are probed. However, as creation dynamics, which adds even further significancewill be seen shortly, lying at the border of experimental to this benchmarking experiment (see also Reiss, 2009).feasibility, the expected pair yields are generally rather A formal treatment of the process has also been given insmall. It is worth mentioning here that lasers can also Ilderton, 2011, where special emphasis is put on effectsbe applied for abundant generation of e+ -e− pairs (Chen stemming from the finite duration of the laser pulse.et al., 2009, 2010). When a solid target is irradiated by an Figure 17 shows a survey of various combinations ofintense laser pulse, a plasma is formed and electrons are incoming electron energies and optical laser intensitiesaccelerated to high energies. They may emit radiation which give rise to an observable pair yield. It covers theby bremsstrahlung which efficiently converts into e+ -e− range from the perturbative few-photon regime (ξ0 ≈ 0.1pairs through the Bethe-Heitler process. The laser field at I0 ≈ 1017 W/cm2 ) to the highly nonperturbative do-plays an indirect role in the pair production here by serv- main (ξ0 ≈ 10 at I0 ≈ 1021 W/cm2 ), where the con-ing solely as a particle accelerator. The prolific amount tributions from thousands of photon absorption chan-of antimatter generated this way may lead to interesting nels need to be included. We note that few-GeV elec-applications in various fields of science (Müller and Kei- tron beams can be produced today using compact laser-tel, 2009). Abundant production of e+ -e− pairs and of plasma accelerators (Leemans et al., 2006) (see also Sec.high-energy photons in the collision of a multipetawatt XII). Future pair creation studies may therefore rely onlaser beam and a solid target has been recently investi- all-optical setups, where a laser-generated electron beamgated in Nakamura et al., 2011 and Ridgers et al., 2012. collides with a counterpropagating laser pulse. AnotherIn particular, in Nakamura et al., 2011 it has been shown all-optical setup for pair creation by a seed electron ex-that almost all the laser pulse energy is converted after posed to two counterpropagating laser pulses was putthe collision into a well collimated high-power gamma- forward in Bell and Kirk, 2008, which will be discussedray flash. Whereas, the numerical simulations in Ridgers in Sec. IX.et al., 2012 indicate that about 35 % of the energy of Pair creation studies could also be conducted as a non-a 10 PW laser pulse after the laser-target interaction is standard application of the 17.5 GeV electron beam atconverted into a gamma-ray burst and that simultane- the upcoming European XFEL beamline at DESY (Euro-ously a pure e+ -e− plasma is produced with a maximum pean XFEL, 2011), which will normally serve to generatepositron density of 1026 m−3 . coherent x-ray pulses. However, in combination with a 31 à 150 Rate ~ 105s-1 B. Pair production in nucleus-laser collisions

Electron Energy @GeVD

à 100 perturbative 50 few-photon à While in electron-laser collisions the contribution of SLAC 20 experiment à reaction (ii) to pair production is in general small, it be- 10 nonperturbative comes accessible to experimental observation when the multiphoton projectile electrons are replaced by heavier particles such 5 à transition to as protons or other nuclei. The two-step production pro- tunneling à 2 cess via multiphoton Compton scattering will then be tunneling 1 strongly suppressed by the large projectile mass. The re- 1017 1018 1019 1020 1021 cent commissioning of the LHC at CERN has stimulated 2 Laser Intensity @Wcm D substantial activities on pair production in combinedFIG. 17 (Color online) Transition from the perturbative to laser and nuclear Coulomb fields, which may be viewed asthe fully nonperturbative regimes of e+ -e− pair creation in a generalization of the well-known Bethe-Heitler processelectron-laser collisions. The laser photon energy is 2.4 eV. to strong fields (multiphoton Bethe-Heitler pair produc-Adapted from Hu et al., 2010. tion). The large Lorentz factors γn of the ultrarelativistic nuclear beams lead to efficient enhancement of the laser parameters in the projectile rest-frame. Indeed, when a proton beam with Lorentz factor γp ≈table-top 10-TW optical laser system, it would also be 3000, as presently available at LHC, collides head-on withvery suitable to probe the various regimes of pair pro- a superintense laser beam of intensity I0 ≈ 1022 W/cm2 ,duction. In particular, the production channel (ii) could the Lorentz-boosted laser field strength approaches thebe investigated by a suitable choice of beam parameters critical value Fcr . This circumstance motivated the first(Hu et al., 2010). calculations of nonperturbative pair production in colli- sions of a relativistic nucleus with a superintense near- Other aspects of pair creation by a high-energy photon optical laser beam (Müller et al., 2003a,b). Smaller pro-and a strong laser field have been investigated in recent jectile Lorentz factors may be sufficient, when ultrastrongyears. In Heinzl et al., 2010a, process (i) was considered XFEL pulses are employed (Avetissian et al., 2003). Thein the case where the laser pulse has finite duration. It calculations were based on an S-matrix treatment and as-was found that the finite pulse duration is imprinted on sumed laser fields of circular polarization. Later on, alsothe spectra of created particles. Pair production by a the case of linear field polarization was studied (Kamińskihigh-energy photon and an ultrashort laser pulse was also et al., 2006; Krajewska et al., 2006; Müller et al., 2004;considered in Tuchin, 2010. Quantum interference effects Sieczka et al., 2006). This case is rendered more involvedcan arise in photon-induced pair creation in a two-mode due to the appearance of generalized Bessel functions,laser field of commensurate frequencies (Narozhny and which are of very high order when ξ0 ≫ 1. The underly-Fofanov, 2000). ing S-matrix element is generally of the form Z In addition, the fundamental process (i) may allow for Sp+ ,σ+ ,p− ,σ− = −ie d4 xΨ†p− ,σ− (x)Vn (r)Ψ−p+ ,−σ+ (x).applications as a novel tool in ultrashort pulse spectrom-etry. A corresponding detection scheme for the char- (44)acterization of short gamma-ray pulses of GeV photons It describes the transition of an electron from thedown to the zeptosecond scale, called Streaking at High negative-energy Volkov state Ψ−p+ ,−σ+ (x) to a positive-Energies with Electrons and Positrons (SHEEP), has energy Volkov state Ψ†p− ,σ− (x), which is mediated by thebeen proposed in Ipp et al., 2011. The basic concept Coulomb potential Vn (r) = Z|e|/r of the projectile nu-of SHEEP is based on e+ -e− pair production in vacuum cleus. An alternative approach to the problem based onby a photon of the test pulse, assisted by an auxiliary the polarization operator in a plane electromagnetic wavecounter-propagating intense laser pulse. In contrast to has been developed in Milstein et al., 2006. It allowsconventional streak imaging, two particles with opposite to obtain total production rates analytically. Both ap-charges, electron and positron, are created in the same proaches rely on the strong-field approximation and in-relative phase within the third streaking pulse that co- clude the laser field exactly to all orders, whereas thepropagates with the test pulse. By measuring simulta- nuclear field is treated at leading order in Zα.neously the energy and momentum of the electrons and Since the high-intensity Bethe-Heitler process has notthe positrons originating from different positions within been observed in experiment yet, in recent years physi-the test pulse, its length and, in principle, even its shape cists have proposed scenarios which may allow to realizecan be reconstructed. The time resolution of SHEEP for the various interaction regimes of the process by present-different classes of tests, streaking and strong pulses can day technology. Few-photon Bethe-Heitler pair produc-range from femtosecond to zeptosecond duration. tion in the perturbative domain could be realized in colli- 32

sions of the LHC proton beam with an XUV pulse of an- known because of the intermediate nature of this param-gular frequency ωX ≈ 100 eV and of moderate intensity eter regime. However, by performing a fitting procedureIX ∼ 1014 W/cm2 (Müller, 2009). Corresponding radi- to numerically obtained results, a total pair productionation sources of table-top dimension are available nowa- rate scaling as m(Zα)2 exp(−3.49/χ0,n) was obtained indays in many laboratories. They are based on HHG from Müller et al., 2009b, which closely resembles √ the tunnel-atomic gas jets or solid surfaces (see Sec. IV.B). The rate ing exponential behaviour m(Zα)2 exp(−2 3/χ0,n ).Re+ -e− of pair creation by two-photon absorption close to While in Eq. (44) the influence of the projectile is de- ⋆the energetic threshold (i.e., ωX = un,− ωX & m, for the scribed by an external Coulomb field, the projectile can ⋆angular frequency ωX of the XUV pulse in the rest-frame also be treated as a quantum particle which allows toof the nucleus) is given by (Milstein et al., 2006) study nuclear recoil effects (Krajewska and Kamiński, ⋆ j+2 2010, 2011; Müller and Müller, 2009). Besides, in laser- 1 ωX nucleus collisions, bound-free pair creation can occur Re+ -e− = 3−j (Zα)2 ξ04 ωX⋆ −1 , (45) 4 m where the electron is created in a deeply bound atomic state of the nucleus. The process was studied first forwith j = 0 for linear polarization and j = 2 for circular circular laser polarization (Matveev et al., 2005; Müllerpolarization. et al., 2003c) and later on also for linear polarization In the quasistatic regime of the process sizable pair (Deneke and Müller, 2008), including contributions fromyields require superintense laser fields from a petawatt the various atomic subshells.source in conjunction with an LHC proton beam (Mülleret al., 2003b; Sieczka et al., 2006). Such experimentswill become feasible when petawatt laser pulses are made C. Pair production in a standing laser waveavailable by high-power devices of table-top size, ratherthan by immobile large-scale facilities as they exist at Purely light-induced pair production can occur whenpresent. A method to enable tunneling pair production two noncopropagating laser waves are superimposed.with more compact multiterawatt laser systems has been The simplest field configuration consists of two coun-proposed in Di Piazza et al., 2009b, 2010b. It relies on terpropagating laser pulses of equal frequency and in-the application of an additional weak XUV field, which tensity. The resulting field is a standing wave which isis superimposed on a powerful optical laser wave. In this inhomogeneous both in space and time and a theoreti-two-color setup, the energy threshold for pair creation cal treatment of the process is very challenging. In or-can be overcome by the absorption of one photon from der to render the problem tractable and since the pro-the high-frequency field and several additional photons duction process mainly occurs where the electric fieldfrom the low-frequency field. As a result, by choosing the component of the background field is stronger than the ⋆XUV frequency ωX such that the parameter δ = (2m − magnetic one (Dittrich and Gies, 2000), in the standard ⋆ωX )/m fulfills the conditions 0 < δ ≪ 1, the tunneling approach the resulting standing light wave is approxi-barrier can be substantially lowered and even controlled. mated by a purely electric field oscillating in time. ThisThe pair production rate in the quasistatic regime for approximation is expected to be justified for a strong0 < δ ≪ 1 depends essentially only on the parameters δ (I0 > 1020 W/cm2 ), optical laser field where the typ-and χ0,n , and on the classical nonlinearity parameter ξX ical spatial scale of the field variation λ0 ∼ 1 µm isof the XUV field. For a circularly polarized strong laser much larger than the pair formation length m/|e|E0 =field it becomes (Di Piazza et al., 2009b) p λC Fcr /E0 ≈ 2.6 × 10−2 µm/ I0 [1020 W/cm2 ] (Ritus, 1985). Note the analogy between the formation length

1 p 2 1 Re+ -e− = √ m(Zα)2 ξX 2 2 χ0,n ζ0 exp − (46) m/|e|E0 for pair production and the tunneling length 64 π 3 ζ0 ltun ∼ Ip /|e|E0 in atomic ionization (see Sec. IV.B), 3/2 with Ip being the ionization potential energy.for ζ0 = χ0,n /2δ √ ≪ 1, to be compared with the usualscaling ∼ (−2 3/χ0,n ) in the absence of the XUV field. Pair production in an oscillating electric field is a A related process is laser-assisted Bethe-Heitler pair generalization of the Schwinger mechanism (Schwinger,creation, where the high-frequency photon energy sat- 1951) to time-dependent fields and it has been considered ⋆isfies ωX > 2m. A pronounced channeling of the e+ -e− by many theoreticians, starting from the seminal workspair due to the forces exerted by the laser field after their Brezin and Itzykson, 1970 and Popov, 1971, 1972 (for acreation was found (Lötstedt et al., 2008, 2009). Multi- comprehensive list of references until 2005, see Salaminphoton Bethe-Heitler pair creation in a two-color laser et al., 2006).wave was investigated in Roshchupkin, 2001. While in the laser-electron and laser-nucleus collisions Analytical formulas for positron energy spectra and of the previous subsections the Doppler boost of the laserangular distributions in the tunneling regime of the pro- parameters due to a highly relativistic Lorentz factorcess were obtained in Kuchiev and Robinson, 2007. For could be exploited, in laser-laser collisions this is not pos-pair production at ξ0 ∼ 1 no analytical expressions are sible so that high field strengths E0 or high frequencies 33

TABLE I Number Ne+ -e− of e+ -e− pairs produced by diﬀer-

ent numbers n of laser pulses, with a total energy W of 10 kJ. The threshold value total energy Wth needed to produce one e+ -e− pair is shown in the third column. The precise collision geometry and the pulse parameters can be found in Bulanov et al., 2010a. Adapted from Bulanov et al., 2010a.

n Ne+ -e− at W = 10 kJ Wth [kJ]

2 < 10−18 40 4 < 10−8 20 8 4.0 10 E0/Fcr 16 1.8 × 103 8FIG. 18 (Color) Number of pairs produced by the x-ray as- 24 4.2 × 106 5.1sisted Schwinger mechanism for two diﬀerent values of the x-ray angular frequency ωX , indicated as ω in the ﬁgure, (blueand black solid lines) and the ratio of these catalyzed pairs tothose produced by the standard Schwinger mechanism (blue pair creation probability can be very large and partiallyand black dashed lines), both as functions of the optical ﬁeld compensate for the exponential suppression (Narozhnystrength in units of Fcr . Adapted from Dunne et al., 2009. et al., 2006). Fine details of pair production in a time-dependent os- cillating electric field are being studied nowadays becauseω0 are required in the laboratory frame. Theoreticians they might serve as characteristic signatures to discrim-are therefore aiming to find ways for enhancing the pair inate the process of interest from potentially strongerproduction probability in order to render the process ob- background processes. For example, it was found thatservable in the foreseeable future. the momentum spectrum of the created particles is highly A first possibility to facilitate the observability of pair sensitive to a subcycle structure of the field (Hebenstreitcreation in a standing optical laser wave is to superim- et al., 2009) and that in the presence of an alternating-pose an x-ray photon (or any other high-frequency com- sign time-dependent electric field, coherent interferenceponent) onto the high-field region (Dunne et al., 2009; effects are observed in the Schwinger mechanism (Akker-Monin and Voloshin, 2010; Schützhold et al., 2008). In mans and Dunne, 2012). The observation in Hebenstreitthis way, the Schwinger mechanism is catalyzed so that et al., 2009 found an elegant mathematical explanationthe usual exponential suppression ∼ exp(−π/Υ0 ) is sig- via the Stokes phenomenon (Dumlu and Dunne, 2010).nificantly lowered. For example, in the limit when the Further effects stemming from the precise shape of thex-ray energy approaches the threshold value 2m, the pair external field were analyzed in Dumlu, 2010 and Dumluproduction rate Re+ -e− becomes and Dunne, 2011. Also, the oscillating dynamics of the

π−2 e+ -e− plasma created by a uniform electric field, in- Re+ -e− ∼ m exp − (47) cluding backreaction effects, was investigated (Apostol, Υ0 2011; Benedetti et al., 2011; Han et al., 2010) (see alsoassuming that the x-ray propagation direction is perpen- Bialynicki-Birula and Rudnicki, 2011 and Kim and Schu-dicular to the electric field vector of the strong optical bert, 2011).field. An overview of the pair production enhancement In addition to pair creation in superstrong laser pulseseffect due to the x-ray assistance is shown in Fig. 18. of low frequency, the process is also extensively discussed Another proposal to enhance the pair yield is the ap- in connection with the upcoming XFEL facilities (see,plication of multiple colliding laser pulses instead of only e.g., Alkofer et al., 2001 and Ringwald, 2001). Here thetwo (Bulanov et al., 2010a). It has been demonstrated question arises as to what extent the spatial field depen-that the threshold laser energy necessary to produce a dence may influence the pair creation process, both insingle pair, decreases when the number of colliding pulses terms of total probabilities and particle momentum dis-is increased. The results are summarized in Table I. Pair tributions. According to Noether’s theorem, pair produc-production exceeds the threshold when eight laser pulses, tion in a time-dependent oscillating electric field occurswith a total energy of 10 kJ, are simultaneously focused with conservation of the total momentum, as well as ofon one spot. Doubling (tripling) the number of pulses the total spin. The problem therefore reduces effectivelyleads to an enhancement by two (six) orders of magni- to a two-level system since the field couples negative andtude. The threshold energy drops from 40 kJ for two positive-energy electron states of same momentum andpulses to 5.1 kJ for 24 pulses, clearly indicating that the spin only. The production process exhibits resonancemultiple-pulse geometry is strongly favorable. Besides, it when the energy gap is an integer multiple of the laser fre-was noticed that the pre-exponential volume factor in the quency, leading to a characteristic Rabi flopping between 34

For another numerical approach to space-time depen-

Maximal production probability

dent problems in quantum field theory, we refer to the review Cheng et al., 2010. Moreover, Schwinger pair pro- duction in a space-time dependent electric field pulse has been treated very recently within the Wigner formalism (Hebenstreit et al., 2011). Here, a self-bunching effect of the created particles in phase space, due to the spa- tiotemporal structure of the pulse, was found. Finally, we mention that, unlike a plane-wave field, a spatially focused laser beam is capable to produce e+ -e− pairs from vacuum and this process has been investigated ω0/m in Narozhny et al., 2004 for different field polarizations. Spontaneous pair production may, in principle, also occurFIG. 19 (Color online) Probability spectrum of e+ -e− pair in a nuclear field for charge numbers Z exceeding a crit-production in two counterpropagating laser pulses, with the ical value Zc , which depends on the nuclear model. Forlaser magnetic ﬁeld included (black triangles) and neglected example, Zc = 173 for a uniformly charged sphere with(red crosses). In the ﬁrst case, the labeling (ℓr -ℓl ) signiﬁes thenumber of absorbed photons from the right-left propagating radius 1.2 × 10−12 cm (Berestetskii et al., 1982). Seewave; in the second case, the peak labels denote the total pho- the reviews Baur et al., 2007 and Zeldovich and Popov,ton number (ℓ). A vanishing initial momentum (i.e., positron 1972 for more detailed information also on e+ -e− pairmomentum) and ξ0 = 1 have been assumed. Adapted from production in heavy ion collisions.Ruf et al., 2009.

D. Spin effects and other fundamental aspects of

the negative and positive-energy Dirac continua (Popov, laser-induced pair creation1971). Due to the electron dressing by the oscillatingfield, the resonant laser frequencies are determined by A particularly interesting aspect of tunneling pair cre-the equation ℓω0 = 2hεi, where hεi is the time-averaged ation is the electron and positron spin-polarization. Inelectron energy in the time-dependent oscillating electric general, pronounced spin signatures in a field-inducedfield. Accordingly, when the particle momentum is var- process may be expected when the background fieldied, several resonances occur corresponding to different strength approaches the critical value Fcr (Kirsebomphoton numbers ℓ. This gives rise to a characteristic ring et al., 2001; Walser et al., 2002). This indeed coincidesstructure in the momentum distribution (Mocken et al., with the condition for a sizable yield of tunneling pair2010). production. Studies of spin effects in pair production by Modifications of these well-established properties of a high-energy photon and a strong laser field were per-the pair creation process, when the spatial field depen- formed in Tsai, 1993 and Ivanov et al., 2005, based ondence and, thus, the laser magnetic-field component are considerations on helicity amplitudes and on the spin-accounted for, have been revealed in Ruf et al., 2009. polarization vector, respectively. Characteristic differ-Utilizing an advanced computer code for solving the cor- ences between fermionic and bosonic particles have beenresponding Dirac equation numerically, it was shown that revealed with respect to pair creation in an oscillatingthe positions of the resonances are shifted, several new electric field (Popov, 1972) and in recent studies of theresonances occur, and the resonance lines are split due to Klein paradox (Cheng et al., 2009b; Krekora et al., 2004;the influence of the spatial field dependence (see Fig. 19). Wagner et al., 2010a). In the latter case it was shownThe basic reason for these effects is that, in contrast to that the existence of a fermionic (bosonic) particle in thea uniform oscillating electric field, the photons in the initial state leads to suppression (enhancement) of thecounterpropagating laser pulses carry momentum along pair production probability due to the different quan-the beam axis. Therefore not only the total number ℓ tum statistics. The enhancement in the bosonic case mayof absorbed laser photons matters, but also how many be even exponential due to an avalanche process (Wag-of them have originated from the laser pulse travelling ner et al., 2010b). Concerning tunneling pair creation into the right and left, respectively. For example, for the combined laser and nuclear Coulomb fields, it has beenmultiphoton order ℓ = 5 two different resonance frequen- shown that the internal spin-polarization vector is pro-cies exist now, corresponding to ℓr = 3, ℓl = 2 on the portional in magnitude to χ0,n and, to leading order,one hand, and to ℓr = 4, ℓl = 1, on the other. Due directed along the transverse momentum component ofto the photon momentum, the former two-level scheme the electron (Di Piazza et al., 2010c). A helicity analy-is also broken into a V -type three-level scheme. This sis of pair production in laser-proton collisions revealedcauses a splitting of the resonance lines, in analogy with that: 1) right-handed leptons are emitted in the labora-the Autler-Townes effect known from atomic physics. tory frame under slightly smaller angles with respect to 35

the proton beam than left-handed ones; 2) the rate of V.A and VIII). Now, at the j th step in which an elec-pair creation of spin-1/2 particles exceeds by almost one tron/positron emits a photon or a photon transforms intoorder of magnitude the corresponding quantity for spin-0 (j) (j) an e+ -e− pair, the initial quantity p0,− or k0,− is con-particles (Müller and Müller, 2011). served and it is distributed over the two final particles (an Other fundamental aspects of laser-induced pair cre- electron/positron and a photon in multiphoton Comp-ation have been recently investigated as well. They com- ton scattering and an e+ -e− pair in multiphoton Breit-prise various kinds of e+ -e− correlations (Fedorov et al., Wheeler pair production). Thus, both resulting particles2006; Krajewska and Kamiński, 2008; Krekora et al., at each step will have a value of their own parameter2005), multiple pair creation (Cheng et al., 2009a), ques- ′(j) ′(j) ′(j) ′(j) χ0 = (p0,− /m)(E0 /Fcr ) or κ0 = (k0,− /m)(E0 /Fcr )tions of locality (Cheng et al., 2008) and vacuum decay smaller than that of the incoming particle. Moreover,times (Labun and Rafelski, 2009), and consistency re- due to the special symmetry of the plane-wave field, thestrictions on the maximum laser field strength to guar- (j) (j) quantities p0,− or k0,− are also rigorously conserved be-antee the validity of the external-field approximation tween two steps (see also Sec. III.A). Then, the avalanche(Gavrilov and Gitman, 2008). (k) (k) ends when the parameters χ0,i and κ0,i at a certain step k are smaller than unity for i ∈ [1, . . . , Nk ], with Nk be- ing the number of particles at that step.IX. QED CASCADES The question arises as to whether other field configura- As it was discussed in the previous section, the E-144 tions exist, where an avalanche process can be efficientlyexperiment at SLAC is the only one, so far, where laser- triggered (see the review Aharonian and Plyasheshnikov,driven multiphoton e+ -e− pair production has been ob- 2003 for the development of QED cascades in matter,served. Considering that about 100 positrons have been photon gas and magnetic field). A positive answer todetected in 22000 shots, each comprising the collision of this question has first been given in Bell and Kirk, 2008:about 107 electrons with the laser beam, the process re- even the presence of a single electron initially at restsults to be rather inefficient. One could attribute this in a standing wave generated by two identical counter-to the relatively low intensity I0 of the laser system of propagating circularly polarized laser fields can prime an1.3 × 1018 W/cm2 (ξ0 ≈ 0.4 as the laser photon en- avalanche process already at field intensities of the orderergy was ω0 = 2.4 eV). However, the extremely high of 1024 W/cm2 . We note that in the presence of a singleenergy ε0 of the electron beam (about 46.6 GeV) en- plane wave the same process would require an intensitysured that that the nonlinear quantum parameter χ0 was of the order of Icr , because for an electron initially at restabout unity (χ0 ≈ 0.3). A recent investigation (Sokolov χ0 = E0 /Fcr . From this point of view, the authors of Bellet al., 2010a) has pointed out in general that in the and Kirk, 2008 explain qualitatively the advantage of em-mentioned setup, i.e., an electron beam colliding with ploying two counterpropagating laser beams by means ofa strong laser pulse, RR effects prevent the development an analogy taken from accelerator physics: a collision be-of a cascade or avalanche process with an efficient, pro- tween two particles in their center-of-momenta is muchlific production of e+ -e− pairs even at much larger laser more efficient than if one of the particles is initially atintensities such that ξ0 ≫ 1. By an avalanche or cas- rest, because much more of the initial energy can becade process we mean here a process in which the in- transferred, for example, to create new particles. Thecoming electrons emit high-energy photons in the laser authors approximated the standing wave by a rotatingfield, which can interact with the field itself generating electric field (see Sec. VIII.C). In such a field and fore+ -e− pairs, which, in turn, emit photons again and so an ultrarelativistic electron the controlling parameter ison (of course a cascade process may also be initiated by χ̃0 = (p⊥ /m)(E0 /Fcr ), where p⊥ is the component of thea photon beam rather than by an electron beam). The electron momentum perpendicular to the electric field.above result has been obtained by numerically integrat- By estimating p⊥ ∼ mξ0 (see also Eq. (2)), one ob-ing the kinetic equations, which describe the evolution of tains χ̃0 ≈ I0 [1024 W/cm2 ]/ω0 [eV] (here ω0 , E0 and I0the electron, the positron and the photon distributions in are the standing-wave’s angular frequency, electric fielda plane-wave background field from a given initial elec- amplitude and intensity). The investigation in Bell andtron distribution, and by accounting for the two basic Kirk, 2008 is based on the analysis of the trajectory ofprocesses that couple these distributions, i.e., multipho- the electron in the rotating electric field including RRton Compton scattering and multiphoton Breit-Wheeler effects via the LL equation. Since the momentum of thepair production. The physical reason why an avalanche electron oscillates around a value of the order of mξ0 , theprocess cannot develop in a single plane-wave field can electron emits high-energy photons efficiently that can inbe understood in the following way. In the ultrarelativis- turn trigger the cascade (see Fig. 20). The possibility oftic case ξ0 ≫ 1, the above-mentioned basic processes in describing the evolution of the electron via its classicala plane-wave field are essentially controlled by the pa- trajectory, can be justified as follows. When an elec-rameters χ0 and κ0 , respectively (see also Secs. III.B, tron interacts with a background electromagnetic field 36

equation was evaluated by employing the total emitted

power calculated quantum mechanically. The stochastic nature of the emission of a photon has been taken into account in Duclous et al., 2011. Analogously, the energy of the emitted photon is chosen randomly following the synchrotron spectral distribution and the momentum of the photon is always chosen to be parallel to that of the emitting electron. By contrast, in the pair production process by a photon, since the photon is not deflected by the laser field, it is assumed that after it has propa- gated one wavelength in the field, it decays into an e+ -e− pair. The main result in Duclous et al., 2011 is that at relatively low intensities of the order of 1023 W/cm2 the pair production rate is increased if the quantum nature of the photon emission is taken into account. The reason is that, due to the stochastic nature of the emission pro- cess, the electron can propagate for an unusually largeFIG. 20 (Color online) The number of e+ -e− pairs (N± ) and distance before emitting. In this way it may gain an un-the number of photons (Nγ ) created by an initial single elec- usually large amount of energy and consequently emit atron in a rotating electric ﬁeld as a function of the ﬁeld inten- high-energy photon, that can be more easily convertedsity. The other plotted quantities are described in Bell and into an e+ -e− pair. Moreover, the discontinuous natureKirk, 2008. From Bell and Kirk, 2008. of the (curvature of the) electron trajectory is shown to slow down the tendency of the electrons to migrate to regions where the electric field vanishes.like that of a laser, quantum effects are essentially of two The intensity threshold of the avalanche process in akinds (Baier et al., 1998): the first one is associated with rotating electric field has also been investigated in Fedo-the quantum nature of the electron motion and the sec- tov et al., 2010. Denoting by tacc the time an electronond one with the recoil undergone by the electron when it needs to reach an energy corresponding to χ0 = 1 start-emits a photon. For an ultrarelativistic electron it can be ing from rest in the given field, by te (tγ ) the electronshown that, while the first kind of quantum effects is neg- (photon) lifetime under photon emission (e+ -e− pair pro-ligible, the second kind is large and has to be taken into duction) and by tesc the time after which the electron es-account (Baier et al., 1998). Thus, the basic assumption capes from the laser field, the authors give the followingis that, since the background laser field is strong, the elec- conditions for the occurrence of the avalanche process:tron is promptly accelerated to ultrarelativistic energies, tacc . te , tγ ≪ tesc . Estimates based on the classicalthe motion between two photon emissions is essentially electron trajectory without including RR effects, lead toclassical and, if necessary, only the emissions have to be the simple condition E0 & αFcr for the avalanche to betreated quantum mechanically by including the photon primed in an optical field. The above estimate corre-recoil. On the other hand, photons are assumed to prop- sponds to an intensity of about 2.5 × 1025 W/cm2 , i.e.,agate in the field along straight lines. one order of magnitude larger than what was found in The model employed in Bell and Kirk, 2008 was im- Bell and Kirk, 2008. However, the main result of Fedo-proved in Kirk et al., 2009 by considering colliding pulsed tov et al., 2010 concerns the limitation, brought aboutfields with finite time duration and a realistic represen- on the maximal laser intensity that can be producedtation for the synchrotron spectrum emitted by a rela- in the discussed field configuration by the starting oftivistic electron. The results in Bell and Kirk, 2008 were the avalanche process. In fact, the energy to acceler-essentially confirmed and numerical simulations with lin- ate the electrons and the positrons participating in theearly polarized beams have shown a general insensitiv- cascade has to come from the background electromag-ity of the cascade development to the polarization of netic field. By assuming an exponential increase of thethe beams. Another interesting finding in Kirk et al., number of electrons and positrons, it is found that al-2009 is that the electrons in the standing wave tend to ready at laser intensities of the order of 1026 W/cm2 themigrate to regions where the electric field vanishes and created electrons and positrons have an energy which ex-then they do not contribute to the pair-production pro- ceeds the initial total energy of the laser beams. Thiscess anymore. In both papers Bell and Kirk, 2008 and hints at the fact that at such intensities the collidingKirk et al., 2009 the emission of radiation by the electron laser beams are completely depleted due to the avalanchewas treated classically, i.e., the electron was supposed process. The results obtained from qualitative estimatesto loose energy and momentum continuously although in Fedotov et al., 2010 have been scrutinized in Elkinain Kirk et al., 2009 the damping force-term in the LL et al., 2011 by means of more realistic numerical meth- 37

ods based on kinetic or cascade equations. In general, a Monte Carlo method. Whereas, the instants of radia-if f∓ (r, p, t) (fγ (r, k, t)) is the electron/positron (pho- tion and pair production have been randomly generated.ton) distribution function (upper and lower sign for elec- In particular, the exponential increase of the number oftron and positron,p respectively) in the phase-space (r, p) e+ -e− pairs and the qualitative estimate, for example, of((r, k)) and ε = m2 + p2 (ω = |k|), their evolution the typical energy of the electron at the moment of thein the presence of a given classical electromagnetic field photon emission carried out in Fedotov et al., 2010 have(E(r, t), B(r, t)) is described by the kinetic equations been confirmed (apart from discrepancies within one or- der of magnitude). ∂ p ∂ ∂ In both papers Elkina et al., 2011 and Fedotov et al., + · ± FL (r, p, t) · f∓ (r, p, t) ∂t ε ∂r ∂p 2010 only the case of a rotating electric field was con- sidered. In Bulanov et al., 2010b it is pointed out that Z = dk wrad (r, p + k → k, t)f∓ (r, p + k, t) the limitation on the maximal laser intensity reachable Z (48) before the cascade is triggered, strongly depends on the − f∓ (r, p, t) dk wrad (r, p → k, t) polarization of the laser beams which create the stand- Z ing wave. The paradigmatic cases of a rotating electric + dk wcre (r, k → p, t)fγ (r, k, t), field and of an oscillating electric field are compared. The estimates presented in Bulanov et al., 2010b for the ∂ k ∂ case of a rotating electric field essentially confirm that + · fγ (r, p, t) ∂t ω ∂r the avalanche starts at laser intensities of the order of 1025 W/cm2 . The main physical reason why the cascade Z = dp wrad (r, p → k, t)[f+ (r, p, t) + f− (r, p, t)] (49) process in a circularly polarized standing wave starts at Z such an intensity is that in a rotating electric field the − fγ (r, k, t) dp wcre (r, k → p, t), electron emits photons with typical energies of the order of 0.29 ω0 γ 3 , i.e., proportional to the cube of the Lorentzwhere FL (r, p, t) = e[E(r, t)+(p/ε)×B(r, t)] and where factor of the emitting electron γ (Bulanov et al., 2010b).wrad (r, p → k, t) (wcre (r, k → p, t)) is the local prob- Whereas, in an oscillating electric field the typical emit-ability per unit time and unit momentum that an elec- ted energy scales as γ 2 , such that in order to radiate atron/positron (photon) with momentum p (k) emits (cre- hard photon with a given energy, a much more energeticates) at the space-time point (t, r) a photon with momen- electron is needed. Hence, the authors of Bulanov et al.,tum k (an e+ -e− pair with the electron/positron having 2010b conclude that in an oscillating electric field RR anda momentum p). It is worth pointing out here a con- quantum effects do not play a fundamental role at lasernection between the development of a QED cascade and intensities smaller than Icr and that avalanche processesthe quantum description of RR. In fact, as has been dis- do not constitute a limitation. It is crucial however, forcussed in Sec. VI, from a quantum point of view, RR cor- the conclusion in Bulanov et al., 2010b that the collisionresponds to the multiple recoils experienced by the elec- of the laser beams occurs in vacuum, i.e., the seed elec-tron in the incoherent emission of many photons. Thus, trons and positrons which would trigger the cascade arein the kinetic approach RR is described by those terms in supposed to be created in the collision itself.Eqs. (48) and (49), which do not involve e+ -e− pair pro- The question of the occurrence of the avalanche for twoduction. In fact, in Elkina et al., 2011 it has been shown colliding linearly polarized pulses has also been addressedin the ultrarelativistic case that the equation of motion in Nerush et al., 2011b, where a detailed description offor the average momentum of the electron distribution, the system under investigation has been provided. Inas derived from Eq. (48), coincides with the LL equation fact, previous models had assumed the background elec-in the classical regime χ0 ≪ 1 (note that pair production tromagnetic field as given, neglecting in this way the fieldis exponentially suppressed at χ0 ≪ 1). generated by the electrons and positrons. The approach In Elkina et al., 2011 the background field is approx- followed in Nerush et al., 2011b exploits the existence ofimated as a uniform, rotating electric field E(t) and two energy scales for the photons: one is that of the ex-f± (r, p, t) → f± (p, t) (fγ (r, k, t) → fγ (k, t)). Analo- ternal laser field and of the plasma fields which is muchgously to Duclous et al., 2011, it is assumed that the smaller than m, and the other is that of the photonsmomentum of the photon emitted in multiphoton Comp- produced by the high-energy electrons which is, by con-ton scattering is parallel to that of the emitting electron trast, much larger than m. The evolution of the low-(positron); in the same way, the momenta of the elec- energy photons is described by means of Maxwell’s equa-tron and positron created in multiphoton Breit-Wheeler tions which are solved with a PIC code, i.e., the photonspair production are assumed to be parallel to that of the are treated as a classical electromagnetic field. Whereas,creating photon. The evolution of the electron, positron the production of hard photons as well as the creationand photon distributions has been investigated by nu- of e+ -e− pairs is described as a stochastic process em-merically integrating the resulting cascade equations via ploying a Monte Carlo method. Unless low-energy ones, 38

X. MUON-ANTIMUON AND PION-ANTIPION PAIR

PRODUCTION y/λ0 The production of e+ -e− pairs in strong laser fields has been discussed in Sec. VIII. In view of the ongo- ing technical progress the question arises as to whether also heavier particles such as muon-antimuon (µ+ -µ− ) or pion-antipion (π + -π − ) pairs can be produced with the y/λ0

emerging near-future laser sources. The production of

µ+ -µ− pairs from vacuum in the tunneling regime ap- pears rather hopeless, though, since the required field needs to be close to Fcr,µ = ̺2µ Fcr = 5.6 × 1020 V/cm, with the ratio ̺µ = mµ /m ≈ 207 between the muon mass mµ and the electron mass m. Even by boosting y/λ0

the effective laser fields with the Lorentz factors (∼ 105 )

of the most energetic electron beams available (Bamber et al., 1999), the value of Fcr,µ seems out of reach. The tunneling production of π + -π − pairs is even more diffi- x/λ0 cult as ̺π = mπ /m ≈ 273. However, µ+ -µ− and π + -π − production can occur in microscopic collision processes in laser-generated or laser-driven plasmas, as well as byFIG. 21 (Color online) Snapshot of the normalized electron few-photon absorption from a high-frequency laser wave.density ρe (part a)), of the normalized photon density ργ (partb)), and of the normalized laser intensity ρl (part c)) 25.5 laserperiods after the two laser counterpropagating beams col-lided. The normalized density of positrons is approximately A. Muon-antimuon and pion-antipion pair production inthe same as that of the electrons. Also, the intensity of each laser-driven collisions in plasmascolliding beam is I0 = 3×1024 W/cm2 and the common wave-length is λ0 = 0.8 µm. Adapted from Nerush et al., 2011b. Energetic particle collisions in a plasma can in prin- ciple drive µ+ -µ− and π + -π − production. The plasma may consist either of electrons and ions or of electrons and positrons. Both kinds of plasmas can be produced by intense laser beams interacting with a solid target. With respect to e+ -e− plasmas, this has been predicted by Liang et al., 1998. As has been mentioned in Sec. VIII, abundant amounts of e+ -e− pairs have been recently pro-hard photons are treated as particles and their evolution duced in this manner at LLNL with pair densities ofis described via a distribution function. It is surpris- the order of 1016 cm−3 (Chen et al., 2009, 2010) anding that by considering a single seed electron initially much higher densities of the order of 1022 cm−3 have beenat rest at a node of the magnetic field of a linearly po- also predicted (Shen and Meyer-ter-Vehn, 2001). Theo-larized standing wave, an avalanche process is observed reticians have therefore started to investigate the prop-in the numerical simulation already for a laser intensity erties and time evolution of relativistic e+ -e− plasmasI0 = 3×1024 W/cm2 and a laser wavelength λ0 = 0.8 µm (Aksenov et al., 2007; Hu and Müller, 2011; Kuznetsova(see Fig. 21). The figure clearly shows the formation of et al., 2008; Kuznetsova and Rafelski, 2012; Mustafa andan overdense e+ -e− plasma in the central region |x| . λ0 . Kämpfer, 2009; Thoma, 2009a,b). In particular, it hasIn the same numerical simulations it is found that after been shown (Kuznetsova et al., 2008; Thoma, 2009a,b)about 20 laser periods almost half of the initial energy of that in an e+ -e− plasma of 10 MeV temperature, µ+ -µ−the laser field has been transferred to the plasma. Dis- pairs, π + -π − pairs as well as neutral π 0 can be createdagreement with the predictions in Bulanov et al., 2010b in e+ -e− collisions. The required energy stems from theis stated to be due to the formation of the avalanche in high-energy tails of the thermal distributions.regions between the nodes of the electric and magnetic Also cold e+ -e− plasmas of high density can be gener-field, where the simplified analysis of the electron motion ated nowadays due to dedicated positron accumulationcarried out in Bulanov et al., 2010b is not valid. Fur- and trapping techniques (Cassidy et al., 2005). Whenther analytical insight into the formation of the cascade such a nonrelativistic low-energy plasma interacts with ahas been reported in Nerush et al., 2011a by analyzing superintense laser field, µ+ -µ− pair production can oc-approximate solutions of the cascade equations in the cur as well (Müller et al., 2006, 2008b). In this case,presence of a rotating electric field. the plasma particles acquire the necessary energy by 39

strong coupling to the external field which drives the elec- In the case of µ+ -µ− pair creation, by considering antrons and positrons into violent collisions. The minimum x-ray photon energy of ω0 = 12 keV and a nuclear rela-laser peak intensity to ignite the reaction e+ e− → µ+ µ− tivistic Lorentz factor of γn = 7000, the photon energy inamounts to about 7 × 1022 W/cm2 at a typical optical the rest-frame of the nucleus amounts to ω0⋆ ≈ 2γn ω0 =laser photon energy of ω0 = 1 eV, corresponding to 168 MeV. The energy gap of 2mµ for µ+ -µ− pair pro-ξ0,min = ̺µ ≈ 207. The rate Re+ e− →µ+ µ− of the pro- duction can thus be overcome by two-photon absorptioncess in the presence of a linearly polarized field reads (Müller et al., 2008a, 2009a). Note that because of pro- s nounced recoil effects, the Lorentz factor which would be 1 α2 2 ξ0,min N+ N− required for two-photon µ+ -µ− production by a projec- Re+ e− →µ+ µ− ≈ 3 2 2 4 1 − 2 , (50) tile electron is much larger: γ & 106 , corresponding to a 2 π m ξ0 ξ0 V currently unavailable electron energy in the TeV range.with the number N± of electrons/positrons and the in- At first sight, e+ -e− and µ+ -µ− pair production interaction volume V , which is determined by the laser fo- combined laser and Coulomb fields seem to be very sim-cal spot size. Equation (50) may be made intuitively ilar processes since the electron and muon only differ bymeaningful by introducing the invariant cross section their mass (and lifetime). In this picture, the corre-σe+ e− →µ+ µ− of µ+ -µ− production in an e+ -e− collision sponding production probabilities would coincide whenin vacuum (Peskin and Schroeder, 1995): the laser field strength and frequency are scaled in accor- r ! dance with the mass ratio ̺µ , i.e., Pµ+ -µ− (E0,µ , ω0,µ ) = 4π α2 4m2µ 2m2µ Pe+ -e− (E0 , ω0 ) for E0,µ = ̺2µ E0 and ω0,µ = ̺µ ω0 . This σe+ e− →µ+ µ− = 1 − ∗ 2 1 + ∗ 2 , (51) simple scaling argument does not apply, however, as the 3 ε∗ 2 ε ε large muon mass is connected with a correspondinglywhere the upper index ∗ indicates quantities in the small Compton wavelength λC,µ = λC /̺µ ≈ 1.86 fmcenter-of-mass system of the colliding electron and (1 fm = 10−13 cm), which is smaller than the radius ofpositron, as, e.g., the common electron and positron en- most nuclei. As a result, while the nucleus can be ap-ergy ε∗ . Now, by exploiting the fact that the quan- proximately taken as pointlike in e+ -e− pair productiontity Re+ -e− /V is a Lorentz invariant and that in the (λC ≈ 386 fm), its finite extension must be taken intopresent physical scenario ε∗ can be estimated as mξ0 , account in µ+ -µ− pair production. Pronounced nuclearEq. (50) implies the usual relation Re∗+ e− →µ+ µ− /V ∗ ∼ size effects have also been found for µ+ -µ− production inσe+ e− →µ+ µ− n∗+ n∗− between the number of events per unit relativistic heavy-ion collisions (Baur et al., 2007).volume and per unit time Re∗+ e− →µ+ µ− /V ∗ and the cross Muon pair creation in XFEL-nucleus collisions can besection σe+ e− →µ+ µ− ; n∗± = N± /V ∗ denote here the par- calculated via the amplitude in Eq. (44), with the nu-ticle densities. The process e+ e− → µ+ µ− in the pres- clear potential Vn (r) arising from an extended nucleus.ence of an intense laser wave has also been considered in It leads to the appearance of a nuclear form factor F (q2 )Nedoreshta et al., 2009. which depends on the recoil momentum q. For example, In laser-produced electron-ion plasmas resulting from F (q2 ) = exp(−q2 a2 /6) for a Gaussian nuclear charge dis-intense laser-solid interactions, µ+ -µ− and π + -π − pairs tribution of root-mean-square radius a. Since the typi-can be generated by the cascade mechanism via energetic cal recoil momentum is q ∼ mµ , the form factor leads tobremsstrahlung, like in the case of e+ -e− pair produc- substantial suppression of the process. The fully differen- (el) (inel)tion mentioned in Sec. VIII. Assuming a laser-generated tial production rate dRµ+ -µ− = dRµ+ -µ− + dRµ+ -µ− mayfew-GeV electron beam, several hundreds to thousands be split into an elastic and an inelastic part, dependingof µ+ -µ− pairs arise from bremsstrahlung conversion in on whether the nucleus remains in its ground state or (el)a high-Z target material (Titov et al., 2009). The pro- gets excited during the process. They read dRµ+ -µ− =duction of π + -π − pairs by laser-accelerated protons was (0) (inel) (0) dRµ+ -µ− Z 2 F 2 (q2 ) and dRµ+ -µ− ≈ dRµ+ -µ− Z[1−F 2 (q2 )],considered in Bychenkov et al., 2001, where a threshold (0)laser intensity of 1021 W/cm2 for the process to occur respectively, with dRµ+ -µ− being the production rate forwas determined. a pointlike proton. Figure 22 shows total µ+ -µ− production rates Rµ⋆ + -µ− in the rest-frame of the nucleus for several nuclei col-B. Muon-antimuon and pion-antipion pair production in liding with an intense XFEL beam. For an extendedhigh-energy XFEL-nucleus collisions nucleus, the elastic rate increases with its charge but de- creases with its size. This interplay leads to the emer- In this subsection another mechanism of µ+ -µ− and gence of maximum elastic rates for medium-heavy ions.π -π − pair creation by laser fields will be pursued, which + Figure 22 also implies that the total rate Rµ+ -µ− =is based on the collision of an x-ray laser beam with an (el) (inel) Rµ+ -µ− + Rµ+ -µ− in the laboratory frame saturates atultrarelativistic nuclear beam. This setup is similar to (inel)the one of Sec. VIII.B. high Z values since Rµ+ -µ− increases with nuclear charge. 40

be triggered. This two-step mechanism thus represents a

combination of the processes considered in Sec. VIII.B and X.A. Besides, it may be considered as a generaliza- tion of the well-established analogy between strong-field Rµ⋆ + -µ− [1/s]

ionization and pair production (see Sec. VIII) to include

also the recollision step.

XI. NUCLEAR PHYSICS

Influencing atomic nuclei with optical laser radiation

is, in general, a difficult task because of the large nuclear Z level spacing ∆E of the order of 1 keV-1 MeV, which + −FIG. 22 Total rates for µ -µ pair creation by two-photon exceeds typical laser photon energies ω0 ∼ 1 eV by ordersabsorption from an intense XFEL beam (ω0 = 12 keV, I0 = of magnitude (Matinyan, 1998). Also the laser’s electric2.5 × 1022 W/cm2 ) colliding with various ultrarelativistic nu- work performed over the tiny nuclear extension rn ∼ 1-5clei (γn = 7000). The triangles show elastic rates, whereas fm is usually too small to cause any sizable effect. In fact,the squares indicate total (“elastic + inelastic”) rates. The the requirement |e|E0 rn ∼ ∆E can only be satisfied for atnumerical data are connected by ﬁt curves. The dotted lineholds for a pointlike nucleus. The production rates are calcu- laser-field amplitudes at least close to Fcr . Direct laser-lated in the rest-frame of the nucleus. Adapted from Müller nucleus interactions have therefore mostly been dismissedet al., 2008a. in the past. On the other hand, laser-induced secondary reactions in nuclei have been explored especially in the late 1990s. Via laser-heated clusters and laser-produced plasmas,For highly-charged nuclei the main contribution stems various nuclear reactions have been ignited, such as fis-from the inelastic channel where the protons inside the sion, fusion and neutron production. In all these cases, (inel)nucleus act incoherently (Rµ+ -µ− ∝ Z). This implies the interaction of the laser field with the target firstthat despite the high charges, high-order Coulomb cor- produces secondary particles such as photo-electrons orrections in Zα are of minor importance. In the collision, bremsstrahlung photons which, in a subsequent step,also e+ -e− pairs are produced by single-photon absorp- trigger the nuclear reaction. For a recent review on thistion in the nuclear field. However, this rather strong subject we refer to Ledingham and Galster, 2010 andbackground process does not deplete the x-ray beam. Ledingham et al., 2003. In XFEL-proton collisions, π + -π − pairs can be gener- In recent years, however, the interest in direct laser-ated as well. A corresponding calculation has been re- nucleus coupling has been revived by the ongoing techno-ported in Dadi and Müller, 2011, which includes both logical progress towards laser sources of increasingly highthe electromagnetic and hadronic pion-proton interac- intensities as well as frequencies. Indeed, when suitabletions. The latter was described approximately by a phe- nuclear isotopes are considered, intense high-frequencynomenological Yukawa potential. It was shown that, fields or superstrong near-optical fields may be capabledespite the larger pion mass, π + -π − pair production of affecting the nuclear structure and dynamics.by two-photon absorption from the XFEL field largelydominates over the corresponding process of µ+ -µ− pair A. Direct laser-nucleus interactionproduction in the Doppler-boosted frequency range ofω0⋆ ≈ 150-210 MeV. This dominance is due to the much 1. Resonant laser-nucleus couplinglarger strength of the strong (hadronic) force comparedwith the electromagnetic force. As a consequence, in this There are several low-lying nuclear transitions in theenergy range µ+ -µ− pairs are predominantly produced keV range, and even a few in the eV range. Examples ofindirectly via two-photon π + -π − production and subse- the latter are 229 Th (∆E ≈ 7.6 eV) and 235 U (∆E ≈ 76quent pion decay, π + → µ+ + νµ and π − → µ− + ν̄µ . eV) (Beck et al., 2007). These isotopes can be excited In relativistic laser-nucleus collisions, µ+ -µ− or π + -π − by the 5th harmonic of a Ti:Sa laser (ω0 = 1.55 eV)pairs can also be produced indirectly within a two-step and by pulses envisaged at the ELI attosecond sourceprocess (Kuchiev, 2007). First, upon the collision, an (ELI, 2011), respectively. Even higher frequencies cane+ -e− pair is created via tunneling pair production. Af- be attained by laser pulse reflection from relativistic fly-terwards, the pair, being still subject to the electromag- ing mirrors of electrons extracted from an underdensenetic forces exerted by the laser field, is driven by the plasma (Bulanov et al., 1994) or possibly also from anfield into an energetic e+ -e− collision. If the collision en- overdense plasma (Habs et al., 2008). Otherwise keV-ergy is large enough, the reaction e+ e− → µ+ µ− may energy photons are generated by XFELs, which, as we 41

been recently produced from resonant betatron motion of 231

Th 5/2− → 5/2+ 186 0.017 25.52 h 1030electrons in a plasma wake (Cipiccia et al., 2011). A new a Estimated via the Einstein A coefficient from τe and ∆Ematerial research center, the Matter-Radiation Interac- b Not listed in the National Nuclear Data Center (NNDC)tions in Extremes (MaRIE) (MaRIE, 2011) is planned, (NNDC, 2011)allowing for both fully coherent XFEL light with photonenergy up to 100 keV and accelerated ion beams. Thephotonuclear pillar of ELI to be set up near Bucharest(Romania) is planned to provide a compact XFEL alongwith an ion accelerator aiming for energies of 4-5 GeV(ELI, 2011). In addition, coherent gamma-rays reach-ing few MeV energies via electron laser interaction are

Nγenvisaged at this facility (ELI, 2011; Habs et al., 2009). With these sources of coherent high-frequency pulses,driving electric dipole (E1) transitions in nuclei is becom-ing feasible (Bürvenich et al., 2006a). Table II displaysa list of nuclei with suitable E1 transitions. Along withan appropriate moderate nucleus acceleration, resonancemay be induced due to the Doppler shift via the factor Z(1 + vn )γn in the counterpropagating setup, with vn andγn being the nucleus velocity and its Lorentz factor, re- FIG. 23 (Color online) Number Nγ of signal photons per nu- cleus per laser pulse for several isotopes with ﬁrst excitedspectively (Bürvenich et al., 2006a). For example, with223 states below 12.4 keV (green squares) and above 12.4 keV Ra and an XFEL frequency of 12.4 keV a factor of (red crosses). The results are plotted versus the atomic(1 + vn )γn = 4 would be sufficient. In general such mod- number Z. The considered European XFEL has a pulseerate pre-accelerations of the nuclei would be of great duration of 100 fs and an average brilliance of 1.6×1025assistance since they increase the number of possible nu- photons/(s mrad2 mm2 0.1% bandwidth) (European XFEL,clear transitions for the limited number of available light 2011). Adapted from Pálﬀy et al., 2008.frequencies. Note that the electric field strength of thelaser pulse transforms analogously in the rest-frame ofthe nucleus, such that the applied laser field in the labo- interactions of laser radiation with nuclei is expected toratory frame may correspondingly be weaker for a coun- pave the way for nuclear quantum optics. Especially con-terpropagating setup. trol in exciting and deexciting certain long-living nuclear For optimal coherence properties of the envisaged high- states would have dramatic implications for nuclear iso-frequency facilities, subsequent pulse applications were mer research (Aprahamian and Sun, 2005; Pálffy et al.,shown to yield notable excitation of nuclei (Bürvenich 2007; Walker and Dracoulis, 1999). As an obvious ap-et al., 2006a). In addition, many low-energy electric plication this would be of relevance for nuclear batter-quadrupole (E2) and magnetic dipole (M1) transitions ies (Aprahamian and Sun, 2005; Walker and Dracoulis,are available. Here it is interesting to note that certain E2 1999), i.e., for controlled pumping and release of energyor M1 transitions can indeed be competitive in strength stored in long-lived nuclear states. In atomic physics, thewith E1 transitions (Pálffy, 2008; Pálffy et al., 2008). STimulated Raman Adiabatic Passage (STIRAP) tech-While the majority of transitions is available for high nique has proven to be highly efficient in controlling pop-frequencies in the MeV domain (requiring substantial nu- ulations robustly with high precision (Bergmann et al.,cleus accelerations), Fig. 23 displays also numerous suit- 1998). On the basis of currently envisaged acceleratorsable nuclear transitions below 12.4 keV along with the and coherent high-frequency laser facilities, it has beenexcitation efficiencies from realistic laser pulses. While recently shown that such an efficient coherent populationindeed experimental challenges are high, resonant direct transfer will also be feasible in nuclei (Liao et al., 2011). 42

Most recently, a nuclear control scheme with optimized muonic wave-function has a large overlap with the nu-pulse shapes and sequences has been developed in Wong cleus. Precise measurements of x-ray transitions be-et al., 2011. tween stationary muonic states are therefore sensitive to Serious challenges are certainly imposed by the nuclear nuclear-structure features such as finite size, deforma-linewidths that may either be too narrow to allow for suf- tion, surface thickness, or polarization.ficient interaction with the applied laser pulses or inhibit When a muonic atom is subjected to a strong laserexcitations and coherences due to large spontaneous de- field, the muon becomes a dynamic nuclear probe whichcay. Decades of research in atomic physics allowing now is periodically driven across the nucleus by the field. Thisfor shaping atomic spectra via quantum interference (Ev- can be inferred, for example, from the high-harmonic ra-ers and Keitel, 2002; Kiffner et al., 2010; Postavaru et al., diation emitted by such systems (Shahbaz et al., 2010,2011) raise hopes that such obstacles may be overcome 2007). Figure 24 compares the HHG spectra from muonicin the near future as well. hydrogen versus muonic deuterium subject to a very Direct photoexcitation of giant dipole resonances strong XUV laser field. Such fields are envisaged at thewith few-MeV photons via laser-electron interaction was ELI attosecond source (see Fig. 1). Due to the differ-shown to be feasible (Weidenmüller, 2011) based on en- ent masses Mn of the respective nuclei (Mn = mp for avisaged experimental facilities such as ELI. Finally, care hydrogen nucleus and Mn ≈ mp +mn for a deuterium nu-has to be taken to compare the laser-induced nuclear cleus by neglecting the binding energy, with mp/n beingchannels with competing nuclear processes via, for ex- the proton/neutron mass), muonic hydrogen gives rise toample, bound electron transitions or electron captures in a significantly larger harmonic cut-off energy. The reasonthe atomic shells (Pálffy, 2010; Pálffy et al., 2007). can be understood by inspection of the ponderomotive energy

e2 E02 e2 E02

1 12. Nonresonant laser-nucleus interactions Up = = + , (52) 4ω02 Mr 4ω02 mµ Mn Already decades ago a lively debate was started on which depends in the present case on the reduced masswhether nuclear β-decay may be significantly affected Mr = mµ Mn /(mµ + Mn ) of the muon-nucleus system.by the presence of a strong laser pulse or not (Akhme- The reduced mass of muonic hydrogen (≈ 93 MeV) isdov, 1983; Becker et al., 1984; Nikishov and Ritus, 1964b; smaller than that of muonic deuterium (≈ 98 MeV) andReiss, 1983) and a conclusive experimental answer to this this implies a larger ponderomotive energy and an en-issue is still to come. Most recently the notion of affect- larged plateau extension. The influence of the nuclearing nuclear α-decay with strong laser pulses has been mass can also be explained by the separated motions ofdiscussed showing that moderate changes of such nuclear the atomic binding partners. The muon and the nucleusreactions with the strongest envisaged laser pulses are are driven by the laser field into opposite directions alongindeed feasible (Castañeda Cortés et al., 2012, 2011). the laser’s polarization axis. Upon recombination their When the laser intensity is high enough (I0 > 1026 kinetic energies sum up as indicated on the right-handW/cm2 ) low-frequency laser fields are able to influence side of Eq. (52). Within this picture, the larger cut-offthe nuclear structure without necessarily inducing nu- energy for muonic hydrogen results from the fact that,clear reactions. In such ultrastrong fields, low-lying nu- due to its smaller mass, the proton is more strongly ac-clear levels get modified by the dynamic (AC-) Stark shift celerated by the laser field than the deuteron.(Bürvenich et al., 2006b). These AC-Stark shifts are of Due to the large muon mass, very high harmonic cut-the same order as in typical atomic quantum optical sys- off energies can be achieved via muonic atoms with chargetems relative to the respective transition frequencies. At number Z in the nonrelativistic regime of interaction.even higher, supercritical intensities (I0 > 1029 W/cm2 ) Since the harmonic-conversion efficiency as well as thethe laser field induces modifications to the proton root- density of muonic atom samples are rather low, it is im-mean-square radius and to the proton density distribu- portant to maximize the radiative signal strength. A siz-tion (Bürvenich et al., 2006b). able HHG signal requires efficient ionization on the one hand, and efficient recombination on the other. The for- mer is guaranteed if the laser’s electric field amplitudeB. Nuclear signatures in laser-driven atomic and molecular E0 lies just below the border of over-barrier ionization,dynamics Mr2 (Zα)3 Muonic atoms represent traditional tools for nuclear E0 . . (53) Qeff 16spectroscopy by employing atomic physics techniques.Due to the large muon mass compared to that of the Here, Qeff = |e|(Z/Mn + 1/mµ )Mr represents an effec-electron, mµ ≈ 207 m, and because of its correspond- tive charge (Reiss, 1979; Shahbaz et al., 2010). Efficientingly small Bohr radius aB,µ = λC,µ /α ≈ 255 fm, the recollision is guaranteed if the magnetic drift along the 43 -9 10 are quite small, however, because of the large difference between the laser photon energy and the nuclear tran- sition energy. Nuclear excitation can also be triggered Harmonic Signal [arb. u.]

with solid targets. One of the main goals of laser-ion starts (Leemans et al., 2006). In fact, the strong interac-acceleration is to create low-cost devices for medical ap- tion comes into play at distances d of the order of d ∼ 1plications, such as for hadron cancer therapy (Combs fm, which for relativistic processes corresponds to ener-et al., 2009). Several regimes have been identified for gies ε ∼ 1/d ∼ 1 GeV. Thus, a laser-based collider is inlaser-ion acceleration (see also the forthcoming review principle suitable for performing particle-physics exper-Macchi et al., 2012). For laser intensities in the range iments. However, in order to initiate high-energy reac-1018 -1021 W/cm2 and for solid targets with a thickness tions with sizable yield, not only GeV energies are re-ranging from a few to tens of micrometers, the so-called quired but also collision luminosities L at least as hightarget-normal-sheath acceleration is the main mecha- as 1026 -1027 cm−2 s−1 . Meanwhile, for the ultimate goalnism (Fuchs et al., 2006). A further laser-ion interac- of being competitive with the next International Lineartion process is the skin-layer ponderomotive acceleration Collider (ILC, 2011), energies on the order of 1 TeV and(Badziak, 2007). By contrast, the radiation-pressure ac- luminosities of the order of 1034 cm−2 s−1 are requiredceleration regime operates when the target thickness is (Ellis and Wilson, 2001).decreased (see Esirkepov et al., 2004 and Macchi et al., The potential of the Laser-Plasma Accelerator (LPA)2009 for the so-called “laser piston” and “light sail” scheme to develop a laser-plasma linear collider is dis-regimes, respectively). In Galow et al., 2011 a chirped cussed in Schroeder et al., 2010. Two LPA regimes areultrastrong laser pulse is applied to proton acceleration analyzed which are distinguished by the relationship be-in a plasma. Chirping of the laser pulse ensures optimal tween the laserpbeam waist size w0 and the plasma fre-phase synchronization of the protons with the laser field quency ωp = 4πnp e2 /m, with np being the plasmaand leads to efficient proton energy gain from the field. density: 1) the quasilinearp regime at large radius of theIn this way, a dense proton beam (with about 107 protons laser beam ωp2 w02 > 2ξ02 / 1 + ξ02 /2 (ξ0 ∼ 1); 2) the bub- √per bunch) of high energy (250 MeV) and good quality ble regime, at ωp w0 . 2 ξ0 (ξ0 > 1). The latter is(energy spread ∼ 1%) can be generated. less suitable for the collider application because the bub- An alternative promising way for particle acceleration ble cavity leads to defocusing for positrons; besides, theis direct laser acceleration in a tightly focused laser beam focusing forces and accelerating forces are not indepen-(Salamin and Keitel, 2002) or in crossed laser beams dently controllable here as both depend on the plasma(Salamin et al., 2003). Especially efficient accelerations density. Whereas, in the quasilinear regime this is pos-(Bochkarev et al., 2011; Gupta et al., 2007; Salamin, sible due to the existence of a second control parameter2006) can be achieved in a radially polarized axicon laser given by the laser beam waist size. In the following,beam (Dorn et al., 2003). For example, the generation we will discuss qualitatively the scaling properties of theof mono-energetic GeV electrons from ionization in a ra- quasilinear regime. For more accurate expressions thedially polarized laser beam is theoretically demonstrated reader is referred to Schroeder et al., 2010.in Salamin, 2007, 2010. A setup for direct laser acceler- In the standard LPA scheme, the electron plasma waveation of protons and bare carbon nuclei is considered in is driven by an intense laser pulse with duration τ0 ofSalamin et al., 2008. It has been shown that laser pulses the order of the plasma wavelength λp = 2π/ωp , whichof 0.1-10 PW can accelerate the nuclei directly to energies accelerates the electrons injected in the plasma wave byin the range required for hadron therapy. Simulations wave breaking (Esarey et al., 2009; Leemans and Esarey,in Galow et al., 2010 and further optimization studies in 2009; Malka et al., 2008). The accelerating field Ep ofHarman et al., 2011 indicate that protons stemming from the plasma wave can be estimated fromlaser-plasma processes can be efficiently post-accelerated mωpemploying single and crossed pulsed laser beams, focused Ep ∼ ∝ n1/2 p . (55) |e|to spot radii of the order of the laser wavelength. Theprotons in the resulting beam have kinetic energies ex- In fact, in the plasma wave, the charge separation occursceeding 200 MeV and small energy spreads of about 1%. on a length scale of the order of λp , producing a sur-The direct-acceleration method has proved to be efficient face charge density σp ∼ |e|np λp and a field Ep ∼ 4πσp ,also for other applications. In Salamin, 2011 it is shown which corresponds to Eq. (55). The number of elec-that 10 keV helium and carbon ions, injected into 1 TW- trons Ne that can be accelerated in a plasma wave ispower crossed laser beams of radial polarization, can be given approximately by the number of charged particlesaccelerated in vacuum to energies of hundreds of keV required to compensate for the laser-excited wakefield,necessary for ion lithography. having a longitudinal component Ek . From the relation Ek ∼ 4πNe |e|/πw02 , it follows that πnpB. Laser-plasma linear collider Ne ∼ ∝ np−1/2 , (56) ωp3 Laser-electron accelerators have already entered the because w0 ωp ∼ 1 in the quasilinear regime. The inter-GeV energy domain where the realm of particle physics action length of a single LPA stage is limited by laser- 45

diffraction effects, dephasing of the electrons with re- conditions each acceleration stage would be powered byspect to the accelerating field, and laser-energy deple- a laser pulse with an energy of 30 J corresponding totion. Laser-diffraction effects can be reduced by employ- an average power of about 0.5 MW at the required rep-ing a plasma channel, and plasma-density tapering can etition rate of 15 kHz. Such high-average powers arebe utilized to prevent dephasing. Therefore, the LPA beyond the performance of present-day lasers. The fu-interaction length will be determined by the energy de- ture hopes for high-average-power lasers are connectedpletion length Ld . We can estimate the latter by equat- with diode-pumped lasers and new amplifier materials.ing the energy spent for accelerating the Ne electrons Further challenges of LPA colliders are their complexity,along Ld (∼ Ne |e|Ep Ld ) to the energy of the laser pulse as they involve plasma channels and density tapering,(∼ E02 πw02 τ0 /8π). Recalling that w0 ωp ∼ 1, this yields and, most importantly, the problem of how to accelerate positrons. ω02 Ld ∼ λp ∝ np−3/2 . (57) ωp2 C. Laser micro-colliderA staging of LPA is required to achieve high current den-sities along with high energies. The electron energy gain We turn now to another scheme for a laser collider,∆εs in a single-stage LPA is which is based on principles quite different than those of the LPA scheme discussed above. In the LPA scheme, ω02 ∆εs ∼ |e|Ep Ld ∼ m ∝ n−1 p . (58) the electron is accelerated due to its synchronous motion ωp2 with the propagating field. In this way the symmetry in the energy exchange process between the electron andTherefore, the number of stages Ns to achieve a the oscillating field is broken, as required by the Lawson-total acceleration energy ε0 is Ns = ε0 /∆εs ∼ Woodward theorem (Lawson, 1979; Woodward, 1947).(ε0 /m)(ωp /ω0 )2 ∝ np . It corresponds to a total collider Another way to exploit the energy gain of the electronlength Lc of in the oscillating laser field is to initiate high-energy pro- ε0 cesses in situ, i.e., inside the laser beam (McDonald and L c ∼ Ns L d ∼ λp ∝ np−1/2 . (59) m Shmakov, 1999). In this case the temporary energy gain of the electron during interaction with a half cycle of theWhen two identical beams each with N particles and laser wave is used to trigger some processes during whichwith horizontal (vertical) transverse beam size σx (σy ) the electron state may change (in particular, the electroncollide with a frequency f , the luminosity L is defined as may annihilate with a positron) and the desired asymme-L = N 2 f /4πσx σy . In LPA N = Ne and f is the laser try in the energy exchange can be achieved. In fact, thisrepetition rate, then approach is widely employed in the nonrelativistic regime 1 1 f via laser-driven recollisions of an ionized electron with its L∼ . (60) parent ion (see Sec. IV.B). 64π ωp σx σy r02 2 The question arises also as to whether the temporaryThe laser energy Ws required in a single stage in energy gain of the electron in the laser beam can also bethe LPA collider is approximately given by Ws ∼ employed in the relativistic regime at ultrahigh energies.m(λp /r0 )(ω0 /ωp )2 at ξ0 ∼ 1 and the total required power As pointed out in Sec. IV, an extension of the establishedPT amounts to PT ∼ Ns Ws f ∼ ε0 f (λp /r0 ). recollision scheme with normal atoms into the relativistic The above estimates show that, although the number regime is hindered by the relativistic drift. However, theNe of electrons in the bunch as well as the single-stage drift will not cause any problem when positronium (Ps)energy gain ∆εs increase at low plasma densities, the atoms are used because its constituent particles, electronaccelerating gradient ∆εs /Ld nevertheless decreases be- and positron, have the same absolute value of the charge-cause the laser depletion length Ld and the overall col- to-mass ratio (see Sec. III.A).lider length Lc increase as well. Limiting the total length A corresponding realization of high-energy e+ -e− rec-of each LPA in a collider to about 100 m will require ollisions in the GeV domain aiming at particle reactionsa plasma density np ∼ 1017 cm−3 to provide a center- has been proposed in Hatsagortsyan et al., 2006. It re-of mass energy of ∼ 1 TeV for electrons, with ∼ 10 lies on (initially nonrelativistic) Ps atoms exposed toGeV energy gain per stage (Schroeder et al., 2010). For super-intense laser pulses. After almost instantaneousa number of electrons per bunch of Ne ∼ 109 , a laser ionization of Ps in the strong laser field, the free electronrepetition rate of 15 kHz and a transverse beam size of and positron oscillate in opposite directions along theabout 10 nm would be required to reach the goal-value laser electric field and experience the same ponderomo-of 1034 cm−2 s−1 for the accelerator luminosity. At the tive drift motion along the laser propagation direction. Inusual condition w0 ∼ 1/ωp ∼ 10 µm, instead, the lu- this way, the particles acquire energy from the field andminosity amounts to L ∼ 1027 cm−2 s−1 . In the above are driven into periodic e+ -e− collisions (Henrich et al., 46

in a strongly laser-driven e+ -e− plasma may be consid-

ered as the coherent counterpart of the incoherent process e+ e− → µ+ µ− discussed in Sec. X.A. Rigorous quantum-electrodynamical calculations for µ+ -µ− pair production in a laser field have been per- formed in Müller et al., 2006, 2008b,c. In agreement with Eq. (61), they enabled the development of a simple- man’s model in which the rate of the laser-driven pro-FIG. 25 (a) In conventional e+ -e− colliders bunches of ac- cess can be expressed via a convolution of the rescat-celerated electrons and positrons are focused to collide head- tering electron wave packet with the field-free cross sec-on-head incoherently, i.e., the bunches collide head-on-head tion σe+ e− →µ+ µ− (see Eq. 51). The latter attains thebut electrons and positrons in the bunch do not. (b) In the (max)recollision-based collider, the electron and positron originat- maximal value σe+ e− →µ+ µ− ∼ α2 λ2C,µ ∼ 10−30 cm2 ating from the same Ps atom may collide head-on-head coher- ε∗ ≈ 260 MeV (Peskin and Schroeder, 1995). However,ently (Henrich et al., 2004). From Hatsagortsyan et al., 2006. when the field driving the Ps atoms is a single laser wave, the e+ -e− recollision times are long and the µ+ -µ− pro- duction process is substantially suppressed by extensive2004). Provided that the applied laser intensity is large wave packet spreading. This obstacle can be overcomeenough, elementary particle reactions like heavy lepton- when two counterpropagating laser beams are employed.pair production can be induced in these recollisions. The The role of the spreading of the electron wave packet incommon center-of-mass energy ε∗ of the electron and the counterpropagating focused laser beams of circular andpositron at the recollision time arises mainly from the linear polarization has been investigated in detail in Liutransversal momentum of the particles and it scales as et al., 2009. The advantage of the circular-polarizationε∗ ∼ mξ0 . A basic particle reaction which could be trig- setup is the focusing of the recolliding electron wavegered in a laser-driven collider is e+ -e− annihilation with packet. However, this advantage is reduced by a spa-production of a µ+ -µ− pair, i.e., e+ e− → µ+ µ− . The tial offset in the e+ -e− collision when the initial coordi-energy threshold for this process in the center-of-mass nate of the Ps atom deviates from the symmetric positionsystem is 2mµ ≈ 210 MeV. It can be reached with a laser between the laser pulses. The latter imposes a severe re-field such that ξ0 ∼ 200, corresponding to laser intensities striction on the Ps gas size along the laser propagationof the order of 1022 W/cm2 , currently within reach. direction. Thus, the linear-polarization setup is prefer- In addition, the proposed recollision-based laser col- able when the offset at the recollision is very small andlider can yield high luminosities compared to conven- the wave packet size at the recollision is within accept-tional laser accelerators. In the latter, bunches of elec- able limits. Results from a Monte Carlo simulation oftrons and positrons are accelerated and brought into the e+ -e− wave-packet dynamics in counterpropagatinghead-on-head collision. However, the particles in the linearly polarized laser pulses are shown in Fig. 26.bunch are distributed randomly such that each micro- The luminosity L and the number of reactionscopic e+ -e− collision is not head-on-head but has a mean events N for the recollision-based collider with counter-impact parameter bi ∼ ab determined by the beam ra- propagating laser pulses can be estimated as:dius ab , characterizing the collision as incoherent (seeFig. 25). Instead, in the recollision-based collider the 1 L ∼NPs τr f, (62)electron and the positron stem from the same Ps atom a3wpwith initial coordinates being confined within the range (max) σe+ e− →µ+ µ−of one Bohr radius aB ≈ 5.3 × 10−9 cm. Since they are N ∼ τr NPs NL , (63)driven coherently by the laser field, they can recollide a3wpwith a mean impact parameter bc ∼ awp of the order of respectively, where NPs is the number of Ps atoms, τrthe electron wave packet size awp (see Fig. 25). Con- the recollision time of the order of the lasers period, NLsequently, the luminosity contains a coherent component the number of laser pulses, and f the laser repetition(Hatsagortsyan et al., 2006): rate. Taking NPs ≈ 108 (Cassidy and Mills, 2005), f = 1

Np (Np − 1) Np Hz and the spatial extension of the e+ -e− pair from Fig. L= + 2 f, (61) 26, one estimates a luminosity of L ∼ 1027 cm−2 s−1 and b2i bc about one µ+ -µ− pair production event every 103 laserwhere Np is the number of particles in the bunch, and f shots at a laser intensity of 4.7 × 1022 W/cm2 .is the bunch repetition frequency. The coherent com- In conclusion, the scheme of the recollision-based laserponent (Np /b2c )f can lead to a substantial luminosity collider allows to realize high-energy and high-luminosityenhancement in the case when the particle number is collisions in a microscopic setup. However, it is not easilylow and the particle’s wave packet spreading is small, scalable to the parameters of the ILC, namely, to TeVNp a2wp < a2b . Note that the reaction Ps → µ+ µ− arising energies and luminosities of the order of 1034 cm−2 s−1 . 47

Electron Positron into a fermion-antifermion pair was calculated in Kurilin, 3 (a)

2 (b) 1 2004, 2009. In all cases, the effect of the laser field was 1 0,9 found to be small. As a general result, the presence of the y [a.u.] 0

2 0,3 The elastic scattering of a muon neutrino and an electron

y [a.u.]

0 0,2 in the presence of a strong laser field has been considered -1 0,1 in Bai et al., 2012 and multiphoton effects in the cross -2

-3 0 section are predicted. -10 -5 0

z [a.u.] 5 10 -10 -5 0 5 10 External fields can also induce decay processes, which z [a.u.] are energetically forbidden otherwise. In Kurilin, 1999,FIG. 26 (Color) The coordinate-space distributions of the the field-induced lepton decay l− → W − + νl waselectron and the positron wave-packets at the recollision time considered. Since the mass of the initial-state parti-in focused counterpropagating pulses along the z direction cle is smaller than the mass of the decay products,with w0 = 10 µm, λ0 = 0.8 µm and with I0 = 4.7 × the process is clearly impossible in vacuum. The pres-1022 W/cm2 (parts (a) and (c)) and I0 = 1.4 × 1023 W/cm2 ence of the field does allow for such an exotic decay,(parts (b) and (d)). The Ps atom is initially located at theorigin. Spatial coordinates are given in “atomic units”, with but the probability Pl− →W − +νl remains exponentially1 a.u. = 0.05 nm. Adapted from Liu et al., 2009. suppressed, i.e., Pl− →W − +νl ∼ exp(−1/χ0,W ), where χ0,W = χ0 (m/mW )3 . Finally, the production of an e+ -e− pair by high-energyXIII. PARTICLE PHYSICS WITHIN AND BEYOND THE neutrino impact on a strong laser pulse has been cal-STANDARD MODEL culated in Tinsley, 2005. The setup is similar to the e+ -e− pair production processes in QED discussed in The sustained progress in laser technology towards Secs. VIII.A and VIII.B. However, as it was shown inhigher and higher field intensities raises the question as Tinsley, 2005, the laser-induced process ν → ν + e+ + e−to what extent ultrastrong laser fields may develop into is extremely unlikely. At a field intensity of abouta useful tool for particle physics beyond QED. Below we I0 = 3 × 1018 W/cm2 , the production length is on thereview theoretical predictions regarding the influence of order of a light year, even for a neutrino energy of 1 PeV.super-intense laser waves on electroweak processes andtheir potential for probing new physics beyond the Stan-dard Model. B. Particle physics beyond the Standard Model

Recently, a lot of attention has been devoted to the

A. Electroweak sector of the Standard Model possibility of employing intense laser sources to test as- pects of physical theories which go even beyond the Stan- The energy scale of weak interactions is set by the dard Model. In Heinzl et al., 2010b, for example, it ismasses of the W ± and Z 0 exchange bosons, mW ≈ mZ ∼ envisaged that effects of the noncommutativity of space-100 GeV. Therefore, the influence of external laser fields, time modify the kinematics of multiphoton Comptoneven if strong on the scale of QED, is generally rather scattering by inducing a nonzero photon mass. We recallsmall. An overview of weak interaction processes in the that in noncommutative quantum field theories operatorspresence of intense electromagnetic fields has been given X µ are associated to spacetime coordinates xµ , which doin Kurilin, 1999. not commute, but rather satisfy the commutation rela- Various weak decay processes in the presence of in- tions [X µ , X ν ] = iΘµν , with Θµν being an antisymmetrictense laser fields have been considered. They can be di- constant tensor (Douglas and Nekrasov, 2001).vided into two classes: 1) laser-assisted processes which On a different side, one of the still open problems ofalso exist in the absence of the field but may be modi- the Standard Model is the so-called strong CP problemfied due to its presence; 2) field-induced processes which (Kim and Carosi, 2010). The nontrivial structure ofcan only proceed when a background field is present, the vacuum, as predicted by Quantum Chromodynam-providing an additional energy reservoir. With respect ics (QCD), allows for the violation within QCD of theto processes from the first category, π → µ + ν and combined symmetry of charge conjugation (C) and par-µ− → e− + ν̄e + νµ have already been examined (Ritus, ity (P). This implies a value for the neutron’s electric1985). Laser-assisted muon decay has also been revis- dipole moment which, however, is already many ordersited recently (Dicus et al., 2009; Farzinnia et al., 2009; of magnitude larger than experimental upper limits. OneNarozhny and Fedotov, 2008). W ± and Z 0 boson decay way of solving this problem was suggested in Peccei and 48

Quinn, 1977 which required the existence of a massive see the standard review Erber, 1966 and the very currentpseudoscalar boson, called axion. The axion has never overview paper Dunne, 2012 for recent progresses in thebeen observed experimentally although some of its prop- field). On the other hand, laser beams deliver fields mucherties can be predicted on theoretical grounds: it should stronger than those employed in the mentioned experi-be electrically neutral and its mass should not exceed ments (the magnetic field strength of a laser beam with1 eV in order of magnitude. Although being electri- the available intensity of 1022 W/cm2 amounts to aboutcally neutral, the axion is predicted to couple to the elec- 6.5×105 T) but in a microscopic space-time region. How-tromagnetic field F µν (x) through a Lagrangian-density ever, it has been first realized in Mendonça, 2007 that en-term visaged ultrahigh intensities at future laser facilities may g compensate for the tiny space-time extension of the laser Laγ (x) = a(x)F µν (x)F̃µν (x), (64) spot region. In Mendonça, 2007 the coupled equations of 4 the electromagnetic field F µν (x) and the axion field a(x)with g being the photon-axion coupling constant and a(x)  ∂ ∂ µ a + m2 a = 1 gF µν F̃ the axion field. µ a µν The photon-axion Lagrangian density in Eq. (64) has 4 (65) µν µν ∂µ F = g(∂µ a)F̃ mainly two implications: 1) the existence of axions in-duces a change in the polarization of a light beam pass- are solved approximately. It is shown that if a probeing through a background electromagnetic field; 2) a laser field propagates through a strong plane-wave field,photon can transform into an axion (and vice versa) the axion field “grows” at the expense mainly of thein the presence of a background electromagnetic field. probe field itself, whose intensity should be observed toThe first prediction has been tested in experiments decrease. Laser powers of the order of 1 PW have al-like the Brookhaven-Fermilab-Rochester-Trieste (BFRT) ready been shown to provide stronger hints for the pres-(Cameron et al., 1993) and the Polarizzazione del Vuoto ence of axions than magnetic-field-based experiments likecon LASer (PVLAS) (PVLAS, 2011), where a linearly the PVLAS. More realistic Gaussian laser beams arepolarized probe laser field crossed a region in which a considered in Döbrich and Gies, 2010 where the start-strong magnetic field was present of 3.25 T and 5 T ing point is also represented by Eq. (65). The sug-at BFRT and at PVLAS, respectively. Testing the sec- gested experimental setup assumes a probe electromag-ond prediction is the aim of the so-called “light shining netic beam with angular frequency ωp,in passing throughthrough a wall” experiments like the Any Light Particle a strong counterpropagating Gaussian beam with angu-Search (ALPS) (ALPS, 2010), the CERN Axion Solar lar frequency ω0,k and another strong Gaussian beamTelescope (CAST) (CAST, 2008) and Gamma to milli- propagating perpendicularly and with angular frequencyeV particle search (GammeV) (GammeV, 2011) (see also ω0,⊥ . By choosing ω0,⊥ = 2ω0,k , it is found that afterthe detailed theoretical analysis in Adler et al., 2008). a photon-axion-photon double conversion, photons areIn the GammeV experiment, for example, the light of a generated with angular frequencies ωp,out = ωp,in ± ω0,k .Nd:YAG laser passes through a region in which a 5 T The amplitudes of these processes are shown to bemagnetic field is present. A mirror is positioned behind peaked at specific values of the axion mass ma,± =that region in order to reflect the laser light. The axions q 2 (1 ± 1)/2. Since the optical photon 2 ωp,in ω0,k + ω0,kwhich would eventually be created in the magnetic-fieldregion pass through the mirror undisturbed and can be energies are of the order of 1 eV, this setup allows for thereconverted to photons by means of a second magnetic investigation of values of the axion mass in this regime.field, activated after the mirror itself. So far these exper- This is very important because such a region of the ax-iments have given negative results. An interesting exper- ion mass is inaccessible to experiments based on strongimental proposal has been put forward in Rabadán et al., magnetic fields, which can probe regions at most in the2006, where the high-energy photon beam delivered by meV range.an XFEL facility has been suggested as a probe beam to In addition to electrically neutral new particles suchtest regions of parameters (like the axion mass ma or the as axions, yet unobserved particles with nonzero chargephoton-axion coupling constant g) which are inaccessible may also exist. The fact that they have so far escapedvia optical laser light. detection implies that they are either very heavy (ren- The perspective for reaching ultrahigh intensities at dering them a target for large-scale accelerator experi-future laser facilities has stimulated new proposals for ments), or that they are light but very weakly charged.employing such fields to elicit the photon-axion interac- In the latter case, these so-called minicharged particles,tion (Gies, 2009). In fact, an advantage of using strong i.e., particles with absolute value of the electric chargeuniform magnetic fields is that they can be kept strong much smaller than |e|, are suitable candidates for laser-for a macroscopically long time (of the order of hours) based searches (Gies, 2009). Let mǫ and Qǫ = ǫe, withand on a macroscopic spatial region (of the order of 1 m) 0 < ǫ ≪ 1, denote the minicharged particle mass and(for QED processes occurring in a strong magnetic field, charge, respectively. Then, the corresponding critical 49

field scale Fcr,ǫ = m2ǫ /|Qǫ | can be much lower than on the polarization of probe beams and on the detectionFcr . As a consequence, vacuum nonlinearities associ- of a typically very low number of signal photons out ofated with minicharged particles may be very pronounced large backgrounds. Similar challenges are envisaged toin an external laser field with intensity much less than detect the presence of light and weakly-interacting hypo-Icr ∼ 1029 W/cm2 . Moreover, even at optical photon thetical particles like axions and minicharged particles.energies ∼ 1 eV the effective Lagrangian approach might The physical properties (mass, coupling constants, etc.)become inappropriate to describe the relevant physics if of such hypothetical particles are, of course, unknown.mǫ . 1 eV (see Sec. VII.A). In Gies et al., 2006, vac- In this respect intense laser fields may be employed hereuum dichroism and birefringence effects due to the ex- to set bounds on physical quantities like the axion massistence of minicharged particles were analyzed when a and, in particular, to scan regions of physical parameters,probe laser beam with ωp > 2mǫ traverses a magnetic which are inaccessible to conventional methods based, forfield. It was shown that polarization measurements in example, on astrophysical observations.this setup would provide much stronger constraints on Different schemes have been proposed to observe e+ -e−minicharged particles in the mass range below 0.1 eV pair production at intensities below the Schwinger limit,than in previous laboratory searches. which seems now to be feasible in the near future, at least from a theoretical point of view. Corresponding studies would complement the results of the pioneering E-144XIV. CONCLUSION AND OUTLOOK experiment and deepen our understanding of the QED vacuum in the presence of extreme electromagnetic fields. The fast development of laser technology has been This is also connected with the recent investigations onpaving the way to employ laser sources for investigating the development of QED cascades in laser-laser collisions.relativistic, quantum electrodynamical, nuclear and high- In addition to being intrinsically interesting, the devel-energy processes. Starting with the lowest required in- opment of QED cascades is expected to set a limit on thetensities, relativistic atomic processes are already within maximal attainable laser intensity. However, the study ofthe reach of available laser systems, while the proposed quantum cascades in intense laser fields has started rela-methods to compensate the deteriorating effects of the tively recently and is still under vivid development. Morerelativistic drift still have to be tested experimentally. advanced analytical and numerical methods are requiredMoreover, a fully consistent theoretical interpretation of in order to describe realistically and quantitatively suchrecent experimental results on correlation effects in rela- a complex system as an electron-positron-photon plasmativistic multielectron tunneling is still missing. in the presence of a strong driving laser field. Concerning the interaction of free electrons with in- Nuclear quantum optics is also a new exciting andtense laser beams, we have seen that experiments have promising field. Since the energy difference between nu-been performed to explore the classical regime. Only clear levels is typically in the multi-keV and MeV range,the E-144 experiment at SLAC has so far been realized high-frequency laser pulses, especially in combinationon multiphoton Compton scattering, although presently with accelerators, are preferable in controlling nuclearavailable lasers and electron beams would allow for prob- dynamics. As pointed out, especially table-top highly co-ing this regime in full detail. We have also pointed out herent x-ray light beams, envisaged for the future, openthat at laser intensities of the order of 1022 -1023 W/cm2 , up perspectives for exciting applications including nu-RR effects come into play at electron energies of the order clear state preparation and nuclear batteries.of a few GeV. It is envisaged that the quantum radiation- Finally, we want to point out that most of the consid-dominated regime, where quantum and RR effects sub- ered processes have not yet been observed or tested ex-stantially alter the electron dynamics, could be one of perimentally. This is in our opinion one of the most chal-the first extreme regimes of light-matter interaction to lenging aspects of upcoming laser physics, not only frombe probed with upcoming petawatt laser facilities. On an experimental point of view, but also from the point ofthe theoretical side, most of the calculations have been view of theoretical methods. Experimentally, the mainperformed by approximating the laser field as a plane reason is that in order to test, for example, nonlinearwave, as the Dirac equation in the presence of a focused quantum electrodynamics or to investigate nuclear quan-background field cannot be solved analytically. Certainly, tum optics, high-energy particle beams (including photonnew methods have to be developed to calculate photon beams) are required to be available in the same labora-spectra including quantum effects and spatio-temporal tory as the strong laser. On the one hand, the combinedfocusing of the laser field in order to be able to quanti- expertise from different experimental physical communi-tatively interpret upcoming experimental results. ties is required to perform such complex but fundamental Nonlinear quantum electrodynamical effects have been experiments. On the other hand, the fast technologicalshown to become observable at future multipetawatt development of laser-plasma accelerators is very promis-laser facilities, as well as at ELI and HiPER. Here the ing and exciting, as this seems the most feasible waymain challenges concern the measurability of tiny effects towards the realization of stable, table-top high-energy 50